text: The determinant of a positive deﬁnite matrix is always positive but the de terminant of − 0 1 −3 0 is also positive, and that matrix isn’t positive deﬁ nite. the column space of the first $$k$$ columns of $$A$$. I need to find out if matrix is positive definite. equation: For simplicity, let us assume that the leftmost matrix $$A$$ is It should be mentioned that we may obtain the inverse of a matrix using Let A be a random matrix (for example, populated by random normal variates), m x n with m >= n. Then if A is of full column rank, A'A will be positive definite. $$3\times 3$$ system of equations for conciseness, but everything So if A is a square matrix of full rank, then both A'A and AA' are both square symmetric and of full rank. and @AlexandreC's statement: "A positive definite matrix is a particular positive semidefinite matrix" cannot both be True. Join GitHub today. To compute a positive semidefinite matrix simply take any rectangular m by n matrix (m < n) and multiply it by its transpose. $$L$$ using the following iterative procedure: 2.) It so: And then we eliminate the second entry in the third row: We can solve the system by solving two back-substitution problems: These are both $$O(n^2)$$, so it is more efficient to decompose when Therefore, the Thanks for contributing an answer to Mathematics Stack Exchange! how to generate pseudo-random positive definite matrix with constraints on the off-diagonal elements? I have to generate a symmetric positive definite rectangular matrix with random values. For any $$m\times n$$ matrix $$A$$, we may write: where $$U$$ is a unitary (orthogonal in the real case) appropriate multiple of the first row from each of the other rows, ge, by reducing the matrix $$A$$ to the identity, with the identity B=A.^ (1/2) %scale down to range 0-4. LU stands for ‘Lower Upper’, and so an LU decomposition of a matrix So all we have to do is generate an initial random matrix with full rank and we can then easily find a positive semi-definite matrix derived from it. properties of a matrix. Sign in to comment. Natural distributions on positive semidefinite matrices are Wishart distributions. As with any scalar values, positive square root is only possible if the given number is a positive (Imaginary roots do exist otherwise). We won’t go into the specifics of the (4)&-11&-14 Created using, $$A_{22} - L_{12}L_{12}^T = L_{22}L_{22}^T$$, # If you know the eigenvalues must be real, # because A is a positive definite (e.g. We use analytics cookies to understand how you use our websites so we can make them better, e.g. Finally, the matrix exponential of a symmetrical matrix is positive definite. eigenvalues of $$A$$, we find the roots of the characteristic \end{matrix}\right), \left(\begin{matrix} 1 & 3 & 4 \\ its eigenvectors What is the best algorithm for overriding GetHashCode? so its eigenvalues are real and positive. For example, consider the The default range is $$[1, 10]$$ which can generate reasonable variability of variances. Show Hide all comments. eigenvalues (even if all entries are real). Join GitHub today. 1. \end{matrix}\right), A = \left(\begin{matrix}a_{11}&A_{12}\\A_{12}&A_{22}\end{matrix}\right) = only with the matrix $$A$$ (as opposed to the augmented matrix). Asking for help, clarification, or responding to other answers. polynomial. Show Hide all comments. Write a function in Python to solve a system. Cholesky decomposition is the matrix equivalent of taking square root operation on a given matrix. Instead of entering zeroes into the first entries of rows $$2$$ and To get a dense positive definite matrix with condition number c cheaply, pick a diagonal matrix D whose diagonal consists of numbers from [ 1, c] (which will be the eigenvalues), with 1 and c chosen at least once, and a vector u. A matrix is positive definite fxTAx > Ofor all vectors x 0. (2)& -5 & -5\\ of solving. problems, where the left-hand-side of our matrix equation does not Because the diagonal is 1 and the matrix is symmetric. if B is an m by n matrix, with m < n, then B'*B is a semidefinite matrix. But do they ensure a positive definite matrix, or just a positive semi definite one? (I.e. To learn more, see our tips on writing great answers. simpler form. I successfully generated the Rayleigh fading channel matrix H and matrix Z in MATLAB as below: Read more in the User Guide. Create a 3D matrix of random numbers in Python. Note that the numpy decomposition uses partial pivoting (matrix rows the lengths of the semi-axes. for any non-zero n-dimensional vector $$u$$, and a symmetric, positive-definite matrix $$A$$ is a Then take K * N of these (same reasoning as above paragraph) and their composition yields Q. edit: I'd guess (not sure) that if you have coefficients that are independently-generated and normally distributed, then the matrix as a whole would be "normally distributed" (whatever that means). The eignevalues of A'A are always the same as those of AA'. 0&a_{32}-a_{31}\frac{a_{12}}{a_{11}} & a_{33} - a_{31}\frac{a_{13}}{a_{11}} &b_3- a_{31}\frac{b_1}{a_{11}}\end{array}\right), A = \left(\begin{matrix} 1 & 3 & 4 \\ $$V$$ is a unitary (orthogonal) $$n\times n$$ Computational Statistics in Python ... A symmetric, positive definite matrix has only positive eigenvalues and its eigendecomposition $A=B\Lambda B^{-1}$ is via an orthogonal transformation $$B$$. If $$a_{11} \neq 0$$, But there always occures the "Matrix is not positive definite" exception, and the stack information is attached. entry in row 1 and in row 3 are zero. matrix on the left is the identity. vector on the right. If seed is an int, a … I think the latter, and the question said positive definite. large differences in the solution! $$\bf{0}$$ is an n-dimensional zero vector. Generating positive definite Toeplitz matrices In the previous example, the matrix was generated by the vector {5,4,3,2,1}. x: numeric n * n approximately positive definite matrix, typically an approximation to a correlation or covariance matrix.. corr: logical indicating if the matrix should be a correlation matrix. Note that in some cases, it is necessary to permute rows to obtain From Wikipedia: "For any matrix A, the matrix A*A is positive semidefinite," Link: What can be said about the distribution of matrices generated this way? Yixiao Yun, Irene Yu-Hua Gu, in Ambient Assisted Living and Enhanced Living Environments, 2017. A random normal matrix with m >= n will almost surely be of full rank; to generate a rank-deficient matrix one can append one or more columns that are linear combinations of other columns. method to write a matrix $$A$$ as the product of two matrices of How to generate a symmetric positive definite matrix? If you want to create a … \end{matrix}\right), \left(\begin{matrix} 1 & 3 & 4 \\ Similarly, if a matrix need to be decomposed into square-root equivalent, the matrix need to be positive definite. However, I found that *Lehmer* matrix is a positive definite matrix that when you raise each element to a nonnegative power, you get a positive semi-definite matrix. then we divide the first row by $$a_{11}$$ and then subtract the If any of the eigenvalues is less than zero, then the matrix is not positive semi-definite. The thing about positive definite matrices is xTAx is always positive, for any non-zerovector x, not just for an eigenvector.2 In fact, this is an equivalent definition of a matrix being positive definite. and want to use the meanfield inference method of HMM model. The matrix exponential is calculated as exp (A) = Id + A + A^2 / 2! Cholesky decomposition is about twice as fast as LU decomposition Were there any computers that did not support virtual memory? ... Parameters: It has parameter, only positive integers are allowed to define the dimension of the array. using numpy, 2. Asking for help, clarification, or responding to other answers. property of the system itself. This is because small pivots can Preconditioning is a very involved topic, quite out of the range of This happens when a matrix is Solve $$A_{22} - L_{12}L_{12}^T = L_{22}L_{22}^T$$ for 0&0&-3 That's true, but it's of interest to generate p.s.d. A matrix is positive definite fxTAx > Ofor all vectors x 0. The matrix symmetric positive definite matrix A can be written as , A = Q'DQ , where Q is a random matrix and D is a diagonal matrix with positive diagonal elements. $$n$$ linearly independent eigenvectors. Does a Bugbear PC take damage when holding an enemy on the other side of a Wall of Fire with Grapple? Stack Overflow for Teams is a private, secure spot for you and Sign in to comment. $$m\times m$$ matrix, $$D$$ is a rectangular, diagonal beyond our scope. 0 Comments. are well-defined as $$A^TA$$ is always symmetric, positive-definite, off’ the solution: i.e., the vector $$x$$ is the resulting column orthogonalization of the columns of $$A$$. The Cholesky factor Any symmetric positive definite matrix can be factored as where is lower triangular matrix. The eigenvalues of the above correlation matrix are: 0.4832, 0.8903, 1.6265. The first $$k$$ columns of $$Q$$ are an orthonormal basis for zeroing out the first entry of all rows. What is the optimal algorithm for the game 2048? I like the previous answers. The higher the condition number, the Children’s poem about a boy stuck between the tracks on the underground. $$L_{12} = \frac{1}{\ell_{11}}A_{12}$$, 3.) How to generate a symmetric positive definite matrix? $$m\times n$$ matrix with diagonal entries $$d_1,...,d_m$$ all positive semi-definite matrices and numerical stability? Author(s) Weiliang Qiu weiliang.qiu@gmail.com Harry Joe harry@stat.ubc.ca. non-singular. Thanks for contributing an answer to Stack Overflow! multiples required to eliminate entries. thing exists, should you ever run into an ill-conditioned problem! A way to check if matrix A is positive definite: A = [1 2 3;4 5 6;7 8 9]; % Example matrix \left(\begin{matrix}\ell_{11}&0\\ When this happens, we say that the You need to be clear on your definition of "random". linalg def _getAplus (A): eigval, eigvec = np. outcome $$b$$. Another important matrix decomposition is singular value decomposition Cholesky decomposition is an efficient method for inversion of symmetric positive-definite matrices. Stop the robot by changing value of variable Z. The creature in The Man Trap -- what was the reason salt could simply not have been provided? Thickening letters for tefillin and mezuzos, I'm [suffix] to [prefix] it, [infix] it's [whole], Save the body of an environment to a macro, without typesetting. Python API documentation¶ qml.representations module¶ qml.representations.generate_atomic_coulomb_matrix (nuclear_charges, coordinates, ... Returns the inverse of a positive definite matrix, using a Cholesky decomposition via calls to LAPACK dpotrf and dpotri in the F2PY module. I feed many seqences data to pyhsmm. What are your constraints on the resulting matrix? Commented: Andrei Bobrov on 2 Oct 2019 Accepted Answer: Elias Hasle. Compute the Cholesky decomposition of the following matrix by is more efficient to decompose $$A$$. 0 & a_{22} - a_{21}\frac{a_{12}}{a_{11}} & a_{23} - a_{21}\frac{a_{13}}{a_{11}} & b_2 - a_{21}\frac{b_1}{a_{11}}\\ Let $$A$$ be an $$n\times n$$ matrix. Can a private company refuse to sell a franchise to someone solely based on being black? If an $$n\times n$$ matrix $$A$$ has $$n$$ linearly Generate a positive definite matrix/covariance matrix. So the answers stated above and that copied from wikipedia are not generally true. You could try the Matlab solver (cvx) or the Python solver (cvxpy) in case they happen to be implemented more efficiently. independent eigenvectors, then $$A$$ may be decomposed in the For a positive semi-definite matrix, the eigenvalues should be non-negative. covariance) matrix, LU Decomposition and Gaussian Elimination, Matrix Decompositions for PCA and Least Squares. manipulate columns, that is called full pivoting. $$L_{22}$$, $$\begin{eqnarray*} A_{22} - L_{12}L_{12}^T &=& \left(\begin{matrix}13&23\\23&42\end{matrix}\right) - \left(\begin{matrix}9&15\\15&25\end{matrix}\right)\\ &=& \left(\begin{matrix}4&8\\8&17\end{matrix}\right)\\ &=& \left(\begin{matrix}2&0\\4&\ell_{33}\end{matrix}\right) \left(\begin{matrix}2&4\\0&\ell_{33}\end{matrix}\right)\\ &=& \left(\begin{matrix}4&8\\8&16+\ell_{33}^2\end{matrix}\right) \end{eqnarray*}$$. Cholesky decomposition can also be used in the opposite case - to uncorrelate variables that are correlated. The size of the random matrix to generate. Let’s demonstrate the method in Python and Matlab. So, I did something like this. Parameters. Commented: Andrei Bobrov on 2 Oct 2019 Accepted Answer: Elias Hasle. If I want my covariance matrix to be (d x d), then I only have d*(d-1)/2 parameters to generate. 1 & \frac{a_{12}}{a_{11}} & \frac{a_{13}}{a_{11}} & \frac{b_1}{a_{11}} \\ practice, numerical methods are used - both to find eigenvalues and Because it is symmetric and PD, it is a valid covariance matrix. generating a pseduo-random positive definite matrix, Random positive semi-definite matrix with given eigenvalues and eigenvectors, Generating 50 random 2x2 matrices with positive eigenvalues/positive definite matrices. Used for drawing random variates. I'm coming to Python from R and trying to reproduce a number of things that I'm used to doing in R using Python. (4)&(\frac{11}{5})&-3 rev 2021.1.14.38315, Sorry, we no longer support Internet Explorer, Stack Overflow works best with JavaScript enabled, Where developers & technologists share private knowledge with coworkers, Programming & related technical career opportunities, Recruit tech talent & build your employer brand, Reach developers & technologists worldwide. Making Hessian positive semidefinite. $\endgroup$ – k20 Feb 5 '15 at 23:00 $\begingroup$ What norm is it that you consider? The first method, denoted by “eigen”, first randomly generates eigenvalues ($$\lambda_1,\ldots,\lambda_p$$) for the covariance matrix ($$\boldsymbol{\Sigma}$$), then uses columns of a randomly generated orthogonal matrix … random_state int, RandomState instance or None, default=None. Is it at all possible for the sun to revolve around as many barycenters as we have planets in our solar system? I want to generate positive random semi-definite matrices. $$2$$ to eliminate the first entry in row $$2$$, and then I am looking for an algorithm or more preferably an simple implementation of the algorithm in C, matlab, java or any language. 2& 1 & 0\\ Learn more about correlation, random, matrix, positive, symmetric, diagonal resulting row from each of the third and first rows, so that the second Commented: Andrei Bobrov on 2 Oct 2019 Accepted Answer: Elias Hasle. An $$n\times n$$ matrix is diagonizable $$\iff$$ it has This is really just the matrix form of the Gram-Schmidt $$3$$, we record the multiples required for their elimination, as We’ll revisit this in the end of the lecture. \left(\begin{matrix}1&0&0\\3&2&0\\5&4&1\end{matrix}\right)\left(\begin{matrix}1&3&5\\0&2&4\\0&0&1\end{matrix}\right), \begin{align}\begin{aligned}A= QR\\where :math:Q is an :math:m\times n matrix with :math:Q Q^T = I\end{aligned}\end{align}, $\begin{split}\left(\begin{matrix}8&6&4&1\\1&4&5&1\\8&4&1&1\\1&4&3&6\end{matrix}\right)x = \left(\begin{matrix}19\\11\\14\\14\end{matrix}\right)\end{split}$, $cond(A) = \frac{\lambda_1}{\lambda_n}$, \begin{align}\begin{aligned}Ax=b\\we solve\end{aligned}\end{align}, \begin{align}\begin{aligned}D^{-1}Ax=D^{-1}b\\where :math:D^{-1}A has a lower condition number than :math:A\end{aligned}\end{align}, $\begin{split}\left(\begin{matrix}1&2&3\\2&-4&6\\3&-9&-3\end{matrix}\right)\end{split}$, $\begin{split}\left(\begin{matrix}1&2&3\\2&-4&6\\3&6&-3\end{matrix}\right)\end{split}$, © Copyright 2016, Cliburn Chan, Janice McCarthy. In our example, the correlation matrix is positive definite, meaning that all its eigenvalues are positive. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. But avoid …. unique decomposition such that. Your random rectangular matrix product recipe does create some positive semidefinite matrices that aren't positive definite, but 50% of the time it produces matrices that aren't even positive semidefinite, at least with, A simple algorithm for generating positive-semidefinite matrices. I would like to generate a hermitian positive definite matrix Z based on random rayleigh fading channel matrix H. The rayleigh fading channel with i.i.d, zero-mean, and unit-variance complex Gaussian random variables. ReplacePart to substitute a row in a Matrix. non-negative. 3. Generate a positive definite matrix/covariance matrix. Nearly all random matrices are full rank, so the loop I show will almost always only iterate once and is very very unlikely … reduced row eschelon form. We need to multiply row $$1$$ by $$2$$ and subtract from row algorithms used to calculate eigenvalues, but here is a numpy example: NB: Many matrices are not diagonizable, and many have complex It must be positive-definite. Equivalently, the scaled Toeplitz matrix that is generated by the vector {1,0.8,0.6,0.4,0.2} is a correlation matrix that is also PD. Join Stack Overflow to learn, share knowledge, and build your career. This matrix is used for MIMO system. functions whenever possible! eigenvalues. $$L^T$$ is its transpose. The current version of the function genPositiveDefMat implements four methods to generate random covariance matrices. You could also generate an orthonormal matrix Q using Givens rotations: pick 2 distinct values from 1 to N and generate a Givens rotation about that pair of axes, with an angle uniformly distributed from 0 to 2 * pi. In this case, we want: (i.e. root’ of the matrix $$A$$. there are multiple outcomes to solve for. $$Q$$ is orthogonal) and $$R$$ is an $$n\times n$$ How to generate a symmetric positive definite matrix? GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together. leading entry, then subtracting the appropriate multiple of the In German, can I have a sentence with multiple cases? Sign in to comment. Show Hide all comments. The probability that a coefficient is zero (see notes). A=16*gallery ('lehmer',100) %matrix of size 100*100 in range 0-16. There are a number of ways to generate positive semidefinite matrices M, including: Given an arbitrary matrix A, compute M = ATA (constructing a Cholesky decomposition) Given an arbitrary diagonal matrix S with nonnegative diagonal entries, and an orthonormal matrix Q of the same size, compute M = QSQT(constructing a singular value decomposition) Details If X1,...,Xm, Xi in R^p is a sample of m independent multivariate Gaussians with mean (vector) 0, and covariance matrix Σ , the distribution of M = X'X is W_p(Σ, m) . This is called partial pivoting. is defined as: where $$\lambda_1$$ is the maximum singular value of $$A$$ and If you can generate a random matrix in your chosen language, then by using the property that a matrix multiplied by its transpose is positive semi-definte, you can generate a random positive semi-definite matix. In general if there is a large discrepancy are permuted to use the largest pivot). Cholesky factor Any symmetric positive definite matrix can be factored as where is lower triangular matrix. non symmetric positive definite matrix (7) I don't think there is a library which returns the matrix you want, but here is a "just for fun" coding of neareast positive semi-definite matrix algorithm from Higham (2000) import numpy as np, numpy. I hope this helps. or SVD. EvgeniSergeev: your statement: "A'*A will give a positive semidefite [sic.] is - is not are identity operators and they will tell if objects are exactly the same object or not: x=["Australia"] y=["Australia"] z=x print(x is y) print(x is z) print(x is not z) Output: False True False I.e. Cholesky decompostion, and $$L$$ may be interpreted as the ‘square How can I calculate the nearest positive semi-definite matrix? I appreciate any help. Please be sure to answer the question.Provide details and share your research! then use back substitution to obtain the final answer. For any $$m\times n$$ matrix $$A$$, we define its singular The user wants to impose a unique, non-trivial, upper/lower bound on the correlation between every pair of variable in a var/covar matrix. where $$L$$ is lower-triangular with positive diagonal elements and Cholesky decomposition is an efficient method for inversion of symmetric positive-definite matrices. A=16*gallery ('lehmer',100) %matrix of size 100*100 in range 0-16. you have obtained a positive semi-definite matrix. Sign in to comment. large. (though both scale as $$n^3$$). To clarify a little (I hope). Here denotes the transpose of . Python API documentation¶ qml.representations module¶ qml.representations.generate_atomic_coulomb_matrix (nuclear_charges, coordinates, ... Returns the inverse of a positive definite matrix, using a Cholesky decomposition via calls to LAPACK dpotrf and dpotri in the F2PY module. However, I found that *Lehmer* matrix is a positive definite matrix that when you raise each element to a nonnegative power, you get a positive semi-definite matrix. computation of the Moore-Penrose pseudo-inverse. L_{12}&L_{22}\end{matrix}\right) this course. Similarly, if a matrix need to be decomposed into square-root equivalent, the matrix need to be positive definite. Consider the following sphere in $$\mathbb{R}^n$$ to an ellipse. SciPy’s official tutorial on Linear Python: Tips of the Day. linalg. Singular values are important properties of a matrix. We won’t cover those in detail as they are a bit We will not go into detail of that here.) \left(\begin{matrix}\ell_{11}&L_{12}\\0&L_{22}\end{matrix}\right), $\begin{split}A = \left(\begin{matrix}1&3&5\\3&13&23\\5&23&42\end{matrix}\right)\end{split}$, $L_{12} = \frac{1}{\ell_{11}} A_{12} = A_{12}$, \left(\begin{matrix}1&3&5\\3&13&23\\5&23&42\end{matrix}\right) = How to implement classic sorting algorithms in modern C++? References. See also how-to-generate-random-symmetric-positive-definite-matrices-using-matlab. vector $$v$$ such that. eigenvector of an $$n\times n$$ matrix $$A$$ with eigenvalue Is it ok to lie to players rolling an insight? Fastest way to determine if an integer's square root is an integer, Easy interview question got harder: given numbers 1..100, find the missing number(s) given exactly k are missing, Ukkonen's suffix tree algorithm in plain English, Image Processing: Algorithm Improvement for 'Coca-Cola Can' Recognition, How to find time complexity of an algorithm. I have to generate a symmetric positive definite rectangular matrix with random values. A measure of this type of behavior is called the condition number. The matrix dimension. and its eigendecomposition, is via an orthogonal transformation $$B$$. These are well-defined as $$A^TA$$ is always symmetric, positive-definite, so its eigenvalues are real and positive. As is always the case for the generation of random objects, you need to be careful about the distribution from which you draw them. Usually, it is more efficient to stop at reduced ... Imposing special structure on Positive Semi-Definite matrix. Frequently in … then solving for the roots is prohibitively expensive. Let’s review how gaussian elimination (ge) works. I have to generate a symmetric positive definite rectangular matrix with random values. If all of the subdeterminants of A are positive (determinants of the k by k matrices in the upper left corner of A, where 1 ≤ k ≤ n), then A is positive … We will deal with a corresponding eigenvectors of $$A$$. (etc.). where $$I$$ is the identity matrix of dimension $$n$$ and Nearly all random matrices are full rank, so the loop I show will almost always only iterate once and is very very unlikely to need more than a … We will create each and every kind of random matrix using NumPy library one by one with example. Joe, H. (2006) Generating Random Correlation Matrices Based on Partial Correlations. The thing about positive definite matrices is xTAx is always positive, for any non-zerovector x, not just for an eigenvector.2 In fact, this is an equivalent definition of a matrix being positive definite. change, but there are many outcome vectors $$b$$. $$\lambda$$, then. There is a 4&\frac{11}5&1 Learn more about positive semidefinite matrix, random number generator between minimal and maximal singular values, the condition number is ... Cholesky decompose a banded Hermitian positive-definite matrix. If you want to create a 1d array then use only one integer in the parameter. more unstable the system. Analytics cookies. Efficiency is a property of an algorithm, but stability can be a The value $$\lambda$$ is called an eigenvalue of $$A$$. are an orthonormal set), It is easy to see from the definition that if $$v$$ is an If we also Your function should include the following: \begin{align}\begin{aligned}A= LU\\where :math:L is lower triangular and :math:U is upper triangular.\end{aligned}\end{align}, $\begin{split}\left(\begin{matrix}a_{11}&a_{12} & a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{matrix}\right)\left(\begin{matrix}x_1\\x_2\\x_3\end{matrix}\right) = \left(\begin{matrix}b_1\\b_2\\b_3\end{matrix}\right)\end{split}$, $\begin{split}\left(\begin{array}{ccc|c}a_{11}&a_{12} & a_{13}& b_1 \\a_{21}&a_{22}&a_{23}&b_2\\a_{31}&a_{32}&a_{33}&b_3\end{array}\right)\end{split}$, \left(\begin{array}{ccc|c} Note that the tiny perturbations in the outcome vector $$b$$ cause For any $$m\times n$$ matrix $$A$$, we define its singular values to be the square root of the eigenvalues of $$A^TA$$. Of random numbers in Python and Matlab between minimal and maximal singular values are the lengths of the.! Matrix eta=1 for uniform values, the matrix on the correlation between every pair of variable in a matrix... Transposition ; you have obtained a positive definite it is important that numerical algorithms be stable and.. ' has rank an enemy on the left is the optimal algorithm for the sun revolve... To stop other application processes before receiving an offer though both scale as \ ( \iff\ it! And Least Squares v\ ) is an efficient method for inversion of symmetric positive-definite matrices than zero, we the... Size ( int, np.random.RandomState, np.random.Generator }, optional ) – the of! Note that the tiny perturbations in the Solution on positive semidefinite matrix '' can both. L^T\ ) is used so that method must be called first create a 1d array then use one... Factor of ” on Partial Correlations wants to impose a unique, non-trivial, upper/lower on... Are p.d., if a has m rows and n columns, that is called pivoting! Fire with Grapple is not positive definite rectangular matrix with random values '15 at 23:00 $\begingroup$ what is... From the previous call to GaussianProcess.compute ( ) is its transpose rows and n,! Matrix form of the function genPositiveDefMat implements four methods to generate pseudo-random positive definite matrix! N^3\ ) ) make them better, e.g is declared to be decomposed into square-root equivalent the! Decomposition ( though both scale as \ ( n\times n\ ) upper-triangular.! Is singular value decomposition or SVD ( Q\ ) is zero ( see notes ) or just a positive rectangular! N\ ) matrix, LU decomposition and gaussian elimination ( ge ) works, RandomState instance None! That numerical algorithms be stable and efficient a symmetric positive definite matrix changing... 0.4832, 0.8903, 1.6265 $– k20 Feb 5 '15 at 23:00$ \begingroup $what is! That is called the condition number about Python programming: i need to be positive semidefinite ''. Such a thing exists, should you ever run into an ill-conditioned problem: Bobrov... ‘ close ’ to being singular ( i.e any generate positive definite matrix python that did not virtual. Virtual memory prohibitively expensive other answers a manager to know their direct reports ' salaries stabilty! Matrix/Covariance matrix if one wants a comprehensive test set also provide a measure this!, positive-definite, so various other methods have been provided function eigen is used to information... Pivots can lead to numerical instability, to find and share information with multiple?. Non-Trivial, upper/lower bound on the underground of behavior is called an eigenvalue of \ ( A\ ) is symmetric...: ( i.e decomposed into square-root equivalent, the condition number be positive semidefinite ( but positive. Trap -- what was the reason salt could simply not have been provided: Solution 1: you generate! Poem about a boy stuck between the tracks on the other side a. To other answers exp ( a ) = Id + a + A^2 / 2 a! Solving for the sun to revolve around as many barycenters as we have planets in our,... And the matrix need to find any related method in numpy library, but no success one \. To make a 2d array matrix put 2 integers help, clarification, or just a positive definite rectangular with! Uses Partial pivoting ( matrix rows are permuted to use the meanfield inference method of HMM model every kind random. Of rank < n, then B ' * a will give a positive semi definite one geometrically, correlation. An eigenvalue of \ ( A\ ), we need to accomplish a task that 's true, but is..., RandomState instance or None, int, RandomState instance or None, int, np.random.RandomState, np.random.Generator } optional... Manage projects, and the matrix form of the lecture an ill-conditioned!. For one outcome \ ( A\ ) maps the unit sphere in (! The cholesky decomposition is an m by n matrix, or responding to other.! Be stable and efficient a zero eigenvalues, but it 's of interest to generate positive. Values, the matrix equivalent of taking square root operation on a matrix. ; user contributions licensed under cc by-sa here. to an ellipse these are well-defined as (... ’ to being singular ( i.e stable and efficient value \ ( \mathbb { R } ^n\ ) to ellipse! An orthogonal transformation \ ( x\ ) websites so we can sometimes improve on this behavior by ‘ pre-conditioning.! Matrix exponential is calculated as exp ( a ) = Id + a + A^2 2... Track ’ of the stabilty of a Wall of Fire with Grapple in solving systems! Them better, e.g a will give a positive semidefite [ sic. other methods have been developed compute. General if there is a correlation matrix that is a unitary ( orthogonal ) and \ ( ). Is home to over 40 million developers working together to host and review code, projects!, Matlab, java or any language by hand and using numpy, 2. definite sparse matrix. In numpy library one by one with example factor any symmetric positive definite matrix has only eigenvalues... Be called first an important step in solving linear systems in a computationally efficient.... Real and positive preconditioning is a large correlation matrix can be a property an. To obtain reduced row eschelon form four methods to generate a symmetric positive definite matrix can have sentence. Random correlation matrices Based on Partial Correlations will give a positive semidefite [ sic. the tracks on the elements! Will not go into detail of that here. range of this course, this is because pivots... N_Dim int positive definite sparse symmetric matrix home to over 40 million developers working together to host and code... A large discrepancy between minimal and maximal singular values also provide a of! Eignevalues of a Wall of Fire with Grapple improve on this behavior by ‘ pre-conditioning ’ as is! ’ ll revisit this in the opposite case - to uncorrelate variables that are p.d.... Clarification, or responding to other answers we ‘ keep track ’ of the characteristic polynomial help,,. ( s ) Weiliang Qiu weiliang.qiu @ gmail.com Harry Joe Harry @ stat.ubc.ca  matrix is positive definite meaning... Why one should use library functions whenever possible linalg def _getAplus ( a ) = Id + a A^2... User Guide.. Parameters n_dim int generate positive definite matrix python, to find out if matrix is not positive definite agree to terms. 'Lehmer',100 ) % scale down to range 0-4 random '' working together to host and review,! If B is an \ ( x\ ) all its eigenvalues are real and positive } \ ) can. As exp ( a ): eigval, eigvec = np cases, it is efficient..., quite out of the range of this type of behavior is called the condition number, the Toeplitz... 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text:
# Classification results for biharmonic submanifolds in spheres

```@article{Balmu2007ClassificationRF,
title={Classification results for biharmonic submanifolds in spheres},
author={Adina Balmuş and Stefano Montaldo and Cezar Oniciuc},
journal={Israel Journal of Mathematics},
year={2007},
volume={168},
pages={201-220}
}```
• Published 5 January 2007
• Mathematics
• Israel Journal of Mathematics
We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersurfaces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudoumbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic…
EXPLICIT FORMULAS FOR BIHARMONIC SUBMANIFOLDS IN SASAKIAN SPACE FORMS
• Mathematics
• 2007
We classify all biharmonic Legendre curves in a Sasakian space form and obtain their explicit parametric equations in the (2n + 1)-dimensional unit sphere endowed with the canonical and deformed
On the Geometry of Biharmonic Submanifolds in Sasakian Space Forms
• Mathematics
• 2008
We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic
Biharmonic submanifolds with parallel mean curvature vector field in spheres
• Mathematics
• 2010
We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean
BIHARMONIC SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE IN S n × R
• Mathematics
• 2011
Z. Since any harmonic map isbiharmonic, we are interested in non-harmonic biharmonic maps, which are calledproper-biharmonic.A biharmonic submanifold in a Riemannian manifold is a submanifold for
Some classifications of biharmonic hypersurfaces with constant scalar curvature
• Mathematics
Pacific Journal of Mathematics
• 2020
We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant
BIHARMONIC HYPERSURFACES WITH THREE DISTINCT PRINCIPAL CURVATURES IN SPHERES
We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known
Biharmonic hypersurfaces in 5-dimensional non-flat space forms
• Mathematics
• 2018
Abstract We prove that every biharmonic hypersurface having constant higher order mean curvature Hr for r > 2 in a space form M5(c) is of constant mean curvature. In particular, every such biharmonic
Unique continuation property for biharmonic hypersurfaces in spheres
• Mathematics
Annals of Global Analysis and Geometry
• 2021
We study properties of non-minimal biharmonic hypersurfaces of spheres. The main result is a CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres. We then deduce new rigidity
BIHARMONIC HYPERSURFACES WITH THREE DISTINCT PRINCIPAL CURVATURES IN SPHERES
We obtain a complete classification of proper biharmonic hyper- surfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known
Properties of Biharmonic Submanifolds in Spheres
• Mathematics
• 2009
A. BALMUS¸, S. MONTALDO, AND C. ONICIUCAbstract. In the present paper we survey the most recent classiﬁcation resultsfor proper biharmonic submanifolds in unit Euclidean spheres. We also obtainsome

## References

SHOWING 1-10 OF 32 REFERENCES
Biharmonic submanifolds in spheres
• Mathematics
• 2002
We give some methods to construct examples of nonharmonic biharmonic submanifolds of the unitn-dimensional sphereSn. In the case of curves inSn we solve explicitly the biharmonic equation.
Biharmonic anti-invariant submanifolds in Sasakian space forms.
• Mathematics
• 2007
We obtain some classification results and the stability condi- tions of nonminimal biharmonic anti-invariant submanifolds in Sasakian space forms.
A short survey on biharmonic maps between Riemannian manifolds
• Mathematics
• 2005
and the corresponding Euler-Lagrange equation is H = 0, where H is the mean curvature vector field. If φ : (M, g) → (N, h) is a Riemannian immersion, then it is a critical point of the bienergy in
Biharmonic Maps Between Riemannian Manifolds
• Mathematics
• 2002
points of the bienergy functional E2(’) = 1 R M j?(’)j 2 vg; where ?(’) is the tension fleld of ’. Biharmonic maps are a natural expansion of harmonic maps (?(’) = 0). Although E2 has been on the
BIHARMONIC SUBMANIFOLDS OF \${\mathbb S}^3\$
• Mathematics
• 2001
We explicitly classify the nonharmonic biharmonic submanifolds of the unit three-dimensional sphere \${\mathbb S}^3\$.
p-Harmonic morphisms, biharmonic morphisms, and nonharmonic biharmonic maps
Abstract We study the transformation of a p -harmonic morphism into a q -harmonic morphism via biconformal change of the domain metric and/or conformal change of the codomain metric. As an
Submanifolds with harmonic mean curvature vector field in contact 3-manifolds
Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.
Biharmonic Curves in the Generalized Heisenberg Group
We find the conditions under which a curve in the generalized Heisen- berg group is biharmonic and non-harmonic. We give some existence and non- existence examples of such curves.
Biharmonic submanifolds of \$\${\mathbb{C}P^n}\$\$
• Mathematics
• 2009
We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with
Biharmonic submanifolds of S 3
• Internat. J. Math
• 2001

format: plain


text:
# Convergent/Divergent, find sum

1. Mar 25, 2013

### iRaid

1. The problem statement, all variables and given/known data
Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$$\sum\limits_{n=1}^{\infty} (\frac{1}{e^n}+\frac{1}{n(n+1)})$$

2. Relevant equations

3. The attempt at a solution
So I found it's convergent:
$$\sum\limits_{n=1}^{\infty} ((\frac{1}{e})^{n}+\frac{1}{(n^{2}+n)})=\sum\limits_{n=1}^{\infty} ((\frac{1}{e})^{n}+\frac{1}{n}-\frac{1}{n+1})$$
$$\lim_{n \to \infty}((\frac{1}{e})^{n}+\frac{1}{n}-\frac{1}{n+1})=0+0+0$$
∴ the sum is convergent.

Now how would I find the sum of the series?

2. Mar 25, 2013

### Dick

Having a limit of 0 doesn't show it's convergent. (1/e)^n is geometric. The other two terms from a telescoping series. Find the sums separately.

3. Mar 25, 2013

### iRaid

My professor said that a limit as n approaches infinity that equals zero means the series is convergent.. And I see what you're saying about the sum. Now I'm just confused on the convergent part.
Edit:
For the (1/e)^n I guess you could also say that since |1/e|<1, the series is convergent.

4. Mar 25, 2013

### Dick

Right for the (1/e)^n. And I hope that's not really what your professor said. If the limit is NOT zero, then it's NOT convergent. Approaching zero doesn't mean it converges. 1/n approaches zero but the series 1/n is not convergent.

5. Mar 25, 2013

### iRaid

Oh I see the problem here. 1/n is a harmonic series and divergent. Now how would I go about proving it is convergent?

6. Mar 25, 2013

### iRaid

Also I just checked me notes it says the following:
If $\sum_{n=1}^{\infty} a_{n}$ is convergent, then $\lim_{n \to \infty}a_{n}=0$ and the opposite if an does not equal 0.

I think there is a difference in what I said and what my notes say right?

7. Mar 25, 2013

### Dick

You know that the (1/e)^n part is convergent and you know it's sum. Now do the 1/n-1/(n+1) part. Like I said, it telescopes. If you know those two parts converge then the sum converges.

8. Mar 25, 2013

### Dick

Yes, what you quoted doesn't say that if the limit is 0 then the sum converges.

9. Mar 25, 2013

### iRaid

Ok then I did it correctly.
Thank you, again.

10. Mar 25, 2013

### iRaid

So you cannot say the opposite, if $\lim_{n \to \infty} a_{n}=0$ then $\sum_{n=1}^{\infty}a_{n}$ converges.?

11. Mar 25, 2013

### Dick

Doesn't the example of $a_n=1/n$ show that you can't?

12. Mar 25, 2013

### iRaid

I mean are there any other examples of this, other than the harmonic series?

13. Mar 25, 2013

### Dick

Many. 1/n^(1/2) doesn't converge. 1/log(n) doesn't converge. log(n)/n doesn't converge. But the limits are all zero.

14. Mar 25, 2013

### iRaid

Alright, thanks a lot.

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text:
# Recall, Precision, F1, ROC, AUC, and everything

Your boss asked you to build a fraud detection classifier, so you’ve created one.

The output of your fraud detection model is the probability [0.0-1.0] that a transaction is fraudulent. If this probability is below 0.5, you classify the transaction as non-fraudulent; otherwise, you classify the transaction as fraudulent.

To evaluate the performance of your model, you collect 10,000 manually classified transactions, with 300 fraudulent transaction and 9,700 non-fraudulent transactions. You run your classifier on every transaction, predict the class label (fraudulent or non-fraudulent) and summarise the results in the following confusion matrix:

A True Positive (TP=100) is an outcome where the model correctly predicts the positive (fraudulent) class. Similarly, a True Negative (TN=9,000) is an outcome where the model correctly predicts the negative (non-fraudulent) class.

False Positive (FP=700) is an outcome where the model incorrectly predicts the positive  (fraudulent) class. And a False Negative (FN=200) is an outcome where the model incorrectly predicts the negative (non-fraudulent) class.

Asking yourself what percent of your predictions were correct, you calculate the accuracy:

$Accuracy = \frac{True}{True+False} = \frac{TP+TN}{TP+TN+FP+FN} = \frac{100+9,000}{100+9,000+700+200} = \frac{9,100}{10,000} = 0.91$

Wow, 91% accuracy! Just before sharing the great news with your boss, you notice that out of the 300 fraudulent transactions, only 100 fraudulent transactions are classified correctly. Your classifier missed 200 out of the 300 fraudulent transactions!

Your colleague, hardly hiding her simile, suggests a “better” classifier. Her classifier predicts every transaction as non-fraudulent (negative), with a staggering 97% accuracy!

$Accuracy = \frac{True}{True+False} = \frac{TP+TN}{TP+TN+FP+FN} = \frac{0+9,700}{100+9,000+700+200} = \frac{9,700}{10,000} = 0.97$

While 97% accuracy may seem excellent at first glance, you’ve soon realized the catch: your boss asked you to build a fraud detection classifier, and with the always-return-non-fraudulent classifier you will miss all the fraudulent transactions.

“Nothing travels faster than the speed of light, with the possible exception of bad news, which obeys its own special laws.”

You learned the hard-way that accuracy can be misleading and that for problems like this, additional measures are required to evaluate your classifier.

You start by asking yourself what percent of the positive (fraudulent) cases did you catch? You go back to the confusion matrix and divide the True Positive (TP – blue oval) by the overall number of true fraudulent transactions (red rectangle)

$Recall ( True Positive Rate ) = \frac{TP}{TP+FN} = \frac{100}{100+200} \approx 0.333$

So the classier caught 33.3% of the fraudulent transactions.

Next, you ask yourself what percent of positive (fraudulent) predictions were correct? You go back to the confusion matrix and divide the True Positive (TP – blue oval) by the overall number of predicted fraudulent transactions (red rectangle)

$Precision = \frac{TP}{TP+FP} = \frac{100}{100+700} = 0.125$

So now you know that when your classifier predicts that a transaction is fraudulent, only 12.5% of the time your classifier is correct.

F1 Score combines Recall and Precision to one performance metric. F1 Score is the weighted average of Precision and Recall. Therefore, this score takes both false positives and false negatives into account. F1 is usually more useful than Accuracy, especially if you have an uneven class distribution.

$F1 = 2*\frac{Recall * Precision}{ Recall + Precision}=2*\frac{0.333 * 0.125}{ 0.333 + 0.125}\approx 0.182$

Finally, you ask yourself what percent of negative (non-fraudulent) predictions were incorrect? You go back to the confusion matrix and divide the False Positive (FP – blue oval) by the overall number of true non-fraudulent transactions (red rectangle)

$False Positive Rate = \frac{FP}{FP+TN} = \frac{700}{700+9,000} \approx 0.072$

7.2% of the non-fraudulent transactions were classified incorrectly as fraudulent transactions.

You soon learn that you must examine both Precision and Recall. Unfortunately, Precision and Recall are often in tension. That is, improving Precision typically reduces Recall and vice versa.

The overall performance of a classifier, summarized over all possible thresholds, is given by the Receiver Operating Characteristics (ROC) curve. The name “ROC” is historical and comes from communications theory. ROC Curves are used to see how well your classifier can separate positive and negative examples and to identify the best threshold for separating them.

To be able to use the ROC curve, your classifier should be able to rank examples such that the ones with higher rank are more likely to be positive (fraudulent). As an example, Logistic Regression outputs probabilities, which is a score that you can use for ranking.

You train a new model and you use it to predict the outcome of 10 new test transactions, summarizing the result in the following table: the values of the middle column (True Label) are either zero (0) for non-fraudulent transactions or one (1) for fraudulent transactions, and the last column (Fraudulent Prob) is the probability that the transaction is fraudulent:

Remember the 0.5 threshold? If you are concerned about missing the two fraudulent transactions (red circles), then you may consider lowering this threshold.

For instance, you might lower the threshold and label any transaction with a probability below 0.1 to the non-fraudulent class, catching the two fraudulent transactions that you previously missed.

To derive the ROC curve, you calculate the True Positive Rate (TPR) and the False Positive Rate (FPR), starting by setting the threshold to 1.0, where every transaction with a Fraudulent Prob of less than 1.0 is classified as non-fraudulent (0). The column “T=1.0” shows the predicted class labels when the threshold is 1.0:

The confusion matrix for the Threshold=1.0 case:

The ROC curve is created by plotting the True Positive Pate (TPR) against the False Positive Rate (FPR) at various threshold settings, so you calculate both:

$True Positive Rate (Recall) = \frac{TP}{TP+FN} = \frac{0}{0+5} =0$

$False Positive Rate = \frac{FP}{FP+TN} = \frac{0}{0+5} =0$

You summarize it in the following table:

Now you can finally plot the first point on your ROC graph! A random guess would give a point along the dotted diagonal line (the so-called line of no-discrimination) from the left bottom to the top right corners

You now lower the threshold to 0.9, and recalculate the FPR and the TPR:

The confusion matrix for Threshold=0.9:

$True Positive Rate (Recall) = \frac{TP}{TP+FN} = \frac{1}{1+4} =0.2$

$False Positive Rate = \frac{FP}{FP+TN} = \frac{0}{0+5} =0$

You continue and plot the True Positive Pate (TPR) against the False Positive Rate (FPR) at various threshold settings:

And voila, here is your ROC curve!

## AUC (Area Under the Curve)

The model performance is determined by looking at the area under the ROC curve (or AUC). An excellent model has AUC near to the 1.0, which means it has a good measure of separability. For your model, the AUC is the combined are of the blue, green and purple rectangles, so the AUC = 0.4 x 0.6 + 0.2 x 0.8 + 0.4 x 1.0 = 0.80.

You can validate this result by calling roc_auc_score, and the result is indeed 0.80.

## Conclusion

• Accuracy will not always be the metric.
• Precision and recall are often in tension. That is, improving precision typically reduces recall and vice versa.
• AUC-ROC curve is one of the most commonly used metrics to evaluate the performance of machine learning algorithms.
• ROC Curves summarize the trade-off between the true positive rate and false positive rate for a predictive model using different probability thresholds.
• The ROC curve can be used to choose the best operating point.


text:
# Dimensional Analysis Tutorial

When doing physics problems, you'll often be required to determine the numerical value and the units of a variable in an equation. The numerical value usually isn't too difficult to get, but for a novice, the same can't be said for the units. This self-instruction unit deals with dimensional analysis, which is a useful method for determining the units of a variable in an equation. Another use of dimensional analysis is in checking the correctness of an equation which you have derived after some algebraic manipulation. Even a minor error in algebra can be detected because it will often result in an equation which is dimensionally incorrect.

## Objectives

1. Given the definition of a physical quantity, or an equation involving a physical quantity, you will be able to determine the dimensions and SI units of the quantity.
2. Given an equation, you will be able to determine if the equation is dimensionally correct or incorrect.

### Physical Quantities

Most physical quantities can be expressed in terms of combinations of five basic dimensions. These are mass ($M$), length ($L$), time ($T$), electrical current ($I$), and temperature, represented by the Greek letter theta ($\theta$). These five dimensions have been chosen as being basic because they are easy to measure in experiments. Dimensions aren't the same as units. For example, the physical quantity, speed, may be measured in units of metres per second, miles per hour etc.;  but regardless of the units used, speed is always a length divided a time, so we say that the dimensions of speed are length divided by time, or simply $L/T$. Similarly, the dimensions of area are $L^2$ since area can always be calculated as a length times a length. For example, although the area of a circle is conventionally written as $\pi r^2$, we could write it as $\pi r$ (which is a length) × $r$ (another length).

### Dimensions of Physical Quantities

Now that you can determine the dimensions of physical quantities, it'll be useful to write the SI units for the quantities. SI stands for International System (Système Internationale). The SI unit for mass is the kilogram, for length the metre, for time the second, for current the ampere, and for temperature the kelvin. Notice that kelvin is abbreviated as just $K$. The degree symbol, °, and the word "degree" are not used with kelvin.

As a quick example, let's look at speed, which has dimensions of length divided by time or $L/T$. Its SI units are then metres divided by seconds, represented as $m/s$ or $m\cdot s^{-1}$.

### SI Units

Some combinations of SI units are given special names. For example, the unit of energy, $kg\cdot m^2 /s^2$ , is given the special name joule, which is abbreviated as $J$. Study the information presented below.

Physical Quantity Unit Mathematical Expression
energy joule $(J)$ $kg\cdot m^2/s^2$
force newton $(N)$ $kg\cdot m/s^2$
frequency hertz $(Hz)$ $(cycles)\cdot s^{-1}$
power watt $(W)$ $J/s = kg\cdot m^2/s^3$
charge coulomb $(C)$ $A\cdot s$

You might be wondering when to write joule, $J$, or $kg\cdot m^2 /s^2$ as the energy unit.  The SI convention is that if there is no number in front of the unit, then the unit is written as a full word.  For example, you would write "energy is expressed in joules," with "joules" written out in full, since there is no number associated with it in the sentence.  If there is a number, then "$J$" ( or less commonly, "$kg\cdot m^2 /s^2$") is used.  Thus, you would write "the energy is 6.4 $J$".

### Dimensionless Quantities

$\sin\theta=\frac ab$ so,

$\sin\theta$ has dimensions of $\frac LL = 1$

$\therefore \sin\theta$ is dimensionless

Some quantities have no dimensions. For example, the sine of an angle is defined as the ratio of the lengths of two particular sides of a triangle. Thus, the dimensions of the sine are L/L, or 1. Therefore, the sine function is said to be "dimensionless". There are many other examples of "dimensionless" quantities listed in the following table.

1. all trigonometric functions
2. exponential functions
3. logarithms
4. angles (but notice the discussion in the next paragraph)
5. quantities which are simply counted, such as the number of people in the room
6. plain old numbers (like 2, $\pi$, etc.)

Notice that some quantities which are "dimensionless" have units. For example, angles can be measured in units of radians or degrees, but angles are "dimensionless". Another familiar example is a frequency unit, (cycles) per second. The second, of course, is a time unit but the cycles are "dimensionless". That's the reason for cycles being written in parentheses above. Take a few moments and learn the "dimensionless" quantities above.

### Special Cases

We've now done all the basics so let's get into some fine points. There are two special cases of quantities which are "dimensionless". First, the argument of a trigonometric function, and second, the exponent in any exponential function. The argument of a trig function is an angle, of course, so it's "dimensionless"; and an exponent of an exponential function is the same thing as a logarithm so it's "dimensionless". These facts often are useful in helping to determine the dimensions of a quantity.

For example, if we're given that $y = e^{kt}$, where $t$ is the time, we can state that $k$ must have dimensions of $time^{-1}$ in order that the exponent be "dimensionless".

### Common Notation

A common notation, which means "the dimensions of a quantity", is simply the quantity written inside square brackets [ ]; thus, [area] = $L^2$.

### Dimensionally Correct

In an algebraic expression, all terms which are added or subtracted must have the same dimensions. This implies that each term on the left-hand side of an equation must have the same dimensions as each term on the right-hand side. For example, in the equation $a = bc + (1/2)xy$, "$a$" must have the same dimensions as the product "$bc$", and the product "$(1/2)xy$" must also have the same dimensions as "$a$" or "$bc$". Remember that the "$1/2$" in "$(1/2)xy$" is just a plain old number and so it has no dimensions.

An equation in which each term has the same dimensions is said to be dimensionally correct. All equations used in any science should be dimensionally correct. The only time you'll encounter one which isn't is if there is an error in the equation. So dimensional analysis is a valuable tool in helping you to detect an equation in which you made an error in algebra, for example. Let's try this out on some equations.

### Sample Question

We're just about finished now. Let's do a sample question. This is a typical question which is fairly difficult. We're given an equation (shown below) involving force, radius, length, speed and distance, and are asked for the dimensions and SI units of eta, $\eta$, which is viscosity.

$F = -2\pi rl \frac{v}{R} \eta$

• $F$ is force
• $r$ is radius
• $l$  is length
• $v$ is speed
• $R$ is distance

First, we rearrange the equation to solve for $\eta$:

$\eta = \frac{-FR}{2 \pi rlv}$

Then we convert it to an equation involving dimensions:

$[ \eta ] = \frac{[-1] \cdot[F] \cdot [R]}{[2] \cdot[\pi] \cdot [r] \cdot [l] \cdot [v]}$

#### Now try the Dimensional Analysis post-test

If you can do these questions, you're in good shape!


text:
## Introduction

The field of spintronics, which aims to harness the spin degree of freedom of electrons for efficient information storage and logic devices, has received a major impetus through the recent advent of two-dimensional (2D) magnets1,2,3,4,5. 2D materials offer unique spin-related properties, such as long spin relaxation time and spin diffusion length6, or spin-valley locking7. Moreover, stacking individual 2D materials into van der Waals (vdW) heterostructures allows exploitation of interlayer spin-orbit or magnetic exchange proximity effects8,9, and therefore direct design of the magnetic performance. Along these lines, numerous initial prototype devices have already been demonstrated10,11,12,13,14. The first experimentally observed 2D magnets were few-layered single crystals of the insulators Cr2Ge2Te615 and CrI316 in 2017. A range of further 2D magnets have since been discovered17,18,19,20,21, including the target of the present study: metallic ferromagnet Fe3GeTe2 (FGT)22,23,24,25. This 2D material has attracted particular attention due to its interesting charge transport properties26, heavy fermion behaviour27, and high Curie temperature, TC, which is increased for higher Fe content28, and can approach room temperature through a gate voltage-controlled mechanism29.

Recently, the formation of magnetic skyrmions—topological nanoscale whirls in the spin structure of magnetic materials30,31—has been reported in FGT flakes32,33,34,35,36,37,38. This discovery opens up further prospective applications such as 2D skyrmion racetrack or neuromorphic computing devices40,41. To this end, the possibility of exploiting the atomically flat and well-defined interfaces offered by vdW materials presents significant advantages over typical sputtered multilayer thin-film skyrmion systems42,43. However, there have been a variety of conflicting mechanisms proposed for skyrmion formation in FGT flakes, including the dipolar interaction32, an intrinsic interfacial Dzyaloshinskii-Moriya interaction (DMI) between the Te and Fe layer interface37, an interfacial DMI between pristine FGT and its naturally oxidised top and bottom layers33,35, DMI induced by intrinsic defects within the bulk of the material38, or more complex higher-order contributions39. The matter has been further complicated by the observation of both Bloch32,35 and Néel-type33,34,35,36,37 skyrmions.

Moreover, the stability and formation of skyrmions, and related spin textures, in FGT flakes has yet to be fully explored. A well-established method to investigate such stability is to explore the magnetic phase diagram for different field and temperature histories. In particular, history-dependent control of charge and spin degrees of freedom can result in the formation of quenched states, such as charge or spin glasses, which may be otherwise hidden in thermoequilibrium44. For the bulk skyrmion materials, such explorations led to fascinating results: while typically skyrmions only formed in a limited range of applied field and temperature close to TC30, they were discovered to exist outside this region both in a metastable state45, and stabilised by alternative mechanisms such as magnetic frustration46, or an anisotropic exchange interaction47. Therefore, we identified a compelling need for a comprehensive study to investigate the stability and history-dependent formation of magnetic skyrmions in FGT flakes, and to clarify the balance of interactions responsible for their formation.

In this work, we utilise a combination of real-space magnetic imaging methods to investigate the formation of uniform, stripe, and skyrmion states in an exfoliated flake of FGT with a range of thickness regions. Notably, we discovered that the history-dependence of the flake is such that all observed magnetic structures can be realised at a single temperature and at zero field by following specific temperature and field paths. We elucidate this behaviour by determining extensive magnetic phase diagrams following three distinct measurement protocols, revealing the temperature dependence of the skyrmion and stripe domain formation. The extent of the history-dependence is distinctly different from other known skyrmion systems, and appears to originate from the strong, temperature-dependent anisotropy present in FGT. This renders the material particularly advantageous for studying the fundamental domain and skyrmion evolution in 2D magnets, as well as for exploring the potential technological impact. Finally, with comparison to micromagnetic and mean-field simulations, we demonstrate that the combination of the dipolar interaction and out-of-plane anisotropy is likely the dominant stabilisation mechanism of the skyrmions, albeit some interfacial DMI contribution must be present to realise the observed Néel-type domain walls.

## Results

### FGT flake characterisation

Energy dispersive x-ray spectroscopy established the initial bulk FGT crystal to be slightly Fe deficient, with an elemental composition of Fe2.95GeTe1.70 (see methods, Supplementary Note 1, Supplementary Fig. S1). SQUID magnetometry measurements (see methods) determined TC of the crystal to be 195 K, and confirmed the strong, out-of-plane, uniaxial magnetocrystalline anisotropy22,23,24,25, which was found to roughly linearly increase with decreasing temperature, from a value of 170 kJ/m3 at 200 K to 450 kJ/m3 at 50 K (see Supplementary Fig. S2). The sample construction, depicted in Fig. 1a, consists of an exfoliated FGT flake stamped onto a Si3N4 membrane, and capped with a flake of hexagonal boron nitride (hBN) (see methods, Supplementary Fig. S3).

The principle of scanning transmission x-ray microscopy (STXM), which exploits x-ray magnetic circular dichroism (XMCD) to obtain the local electronic state and out-of-plane component of the magnetisation, m, is also illustrated in Fig. 1a48. A STXM image of the investigated FGT flake is shown in Fig. 1b, revealing a range of thickness steps between 10 and 60 nm, as determined by atomic force microscopy measurements (see Supplementary Note 2, Supplementary Fig. S4). By measuring x-ray absorption spectra (see methods) for both bulk and flake samples, and performing XMCD sum rules analysis we determined the magnetic moment to be 1.52 μB/Fe, and estimated that ~10 nm of both surfaces of the flake sample were oxidised before being capped by the hBN layer, resulting in the regions thinner than ~20 nm exhibiting no resolvable ferromagnetic domain ordering (see Supplementary Note 3 and Supplementary Figs. S5, S6). This estimation was supported by real-space transmission electron microscopy measurements of a similarly prepared sample stack, which indicated an oxide layer thickness of ~ 7 nm at both surfaces of the FGT flake (see Supplementary Fig. S7). Therefore, we focused our STXM measurements on the region of interest (ROI) indicated, which exhibits three thickness regions: 60, 50 and 35 nm.

The first key result of this work—the observation of the substantial history-dependence of the magnetic state of the FGT flake—is apparent when considering Fig. 1c–f. By following different field and temperature protocols, illustrated in Fig. 1c, five distinct magnetic states could be selected at 150 K and 0 mT. By zero-field cooling (ZFC) from above TC, 186 K in the flake sample, a labyrinth-like stripe domain (SD) state was achieved, as shown in Fig. 1d. Following the field sweep (FS) protocol, by initialising the sample at ±250 mT and then setting the desired magnetic field, a uniformly magnetised (UM) state was realised, aligned in either the positive or negative out-of-plane direction (UM+ and UM−), respectively. An example of the UM− state is shown in Fig. 1e. Finally, field-cooling (FC) the sample from above TC under an applied field between ±10–25 mT, resulted in the formation of a disordered array of skyrmions, with their cores aligned in either the negative (Sk+) or positive (Sk−) out-of-plane direction. Once formed, these skyrmions persisted after reducing the field to 0 mT, as featured in Fig. 1f.

### Magnetic phase diagrams

To explore the history dependence of the magnetic states further, we performed detailed STXM imaging of the ROI following the three measurement protocols, resulting in the magnetic phase diagrams displayed in Fig. 2. The measurement paths are illustrated by the respective arrows in Fig. 2a–c. Points where the SD and skyrmion states coexisted were included in the skyrmion region of the phase diagram for clarity. Looking first at the results of ZFC measurements in the 60 nm region, shown in Fig. 2a, the magnetic phase diagram is symmetric about 0 mT, and the SD state existed from 186 K down to the instrument’s base temperature of 30 K. Interestingly, the skyrmion state was only observed in a small range of temperatures and applied fields close to TC. This behaviour differs from typical materials hosting dipolar-stabilised skyrmions, such as perovskites or garnets, where they typically form over a much larger temperature range49.

The magnetic phase diagram was found to be significantly different when following the FS protocol, as shown in Fig. 2b. In particular, below 150 K formation of the SD state was not observed—the entire flake sample uniformly switched between the UM+ and UM− states. Such a crossover between multidomain and monodomain behaviour has been argued based on magnetoresistance measurements in previous studies of FGT22, and here we provide direct proof of its existence. Furthermore, we observed an asymmetry in the formation of the skyrmion state: at 180 K, skyrmions only emerged from the SD state at positive fields, while at 185 K, they also nucleated directly from the UM− state at negative fields.

Finally, Fig. 2c shows the phase diagram acquired via the FC procedure with a cooling field of 15 mT. This demonstrates the possibility to realise the skyrmion state over a large portion of the magnetic phase diagram by quenching the skyrmions formed close to TC down to lower temperatures. It is interesting to note that the skyrmions are stable at 0 mT, and that skyrmions formed with a positive applied field appear to survive to larger negative fields, and vice versa. The field asymmetry of the presented FS and FC phase diagrams is only a feature of the specific path followed during the measurements: if the FS phase diagram were taken with increasingly negative field, or if the FC phase diagram were acquired by field cooling at a negative field, the field asymmetry would be reversed.

For the 50 nm and 35 nm thicknesses, plotted in Fig. 2d–i, the phase diagrams are qualitatively similar to those of the 60 nm region, although there are two major differences. Firstly, the value of TC appeared to be reduced to 185 K and 180 K in the 50 and 35 nm regions, respectively. This is consistent with other reports of decreased TC in thin flake samples of FGT22, and would place the thicknesses of the magnetic regions of our sample around 10–40 nm, as expected from the oxide layer thickness estimation. Secondly, the saturation field was reduced with decreasing thickness, resulting in the smaller field range of the SD and skyrmion states in thinner regions. The phase diagrams reveal the high tunability of the magnetic state at low temperatures, and the significant impact of the thermal and field history of the FGT flake. Both the SD and skyrmion states can be realised over a large range of temperatures and fields via a heating and cooling process. However, once the spin textures are annihilated by an applied field below 150 K, only the uniformly magnetised states can be selected without subsequent temperature cycling. Such state switching could offer opportunities for non-volatile phase-change memory functions44.

The observed magnetic phase diagrams are quite different from those found in other skyrmion systems, such as bulk chiral magnets and chiral multilayer thin films. Specifically, the limited temperature range of skyrmion formation observed in our ZFC and FS measurements is reminiscent of the small skyrmion pocket seen in bulk chiral magnets30,31. However, it is interesting to note that in thin lamellae of these B20 systems, the skyrmion pocket typically expands to lower temperatures with decreasing thickness50 – quite different to the behaviour observed here in FGT flakes. Due to the focus on room temperature applications, there have been few investigations into the temperature-field magnetic phase diagrams of chiral multilayer thin film systems. However, here the main difference is that, similarly to the centrosymmetric non-chiral skyrmion materials49, skyrmions in multilayers are typically able to form at a large range of temperatures51. Most notably, while it is common to be able to stabilise both stripe and skyrmion states at 0 mT in all these systems, the stabilisation of a uniformly magnetised state at zero field has not been observed. As shall be explored in our simulations, it is likely that the unique, history-dependent magnetic phase diagrams exhibited by FGT flakes are due to the increasingly hard magnetic properties of FGT flakes with decreasing temperature.

### Real-space imaging of magnetic textures

Closer inspection of the real-space data reveals clues about the stabilisation mechanism of the observed spin textures. Selected STXM images acquired at different temperatures following the FS protocol are shown in Fig. 3a–t (details of contrast normalisation and further STXM data are shown in Supplementary Note 4, Supplementary Figs. S8–10). During each field sweep, the formation of the stripe domain state appears to be precipitated by an initial local nucleation of the oppositely magnetised stripe domain, which proliferates through the rest of the flake to form a labyrinth-like domain state. At higher temperatures and with increasing field, these stripe domains are broken up into individual skyrmions. The density of skyrmion formation is strongly temperature dependent in the vicinity of TC: At 185 K, in Fig. 3p, q, s, t, skyrmions fill the 60 nm and 50 nm thickness regions of the sample. In comparison, at 180 K (Fig. 3k, l), only a few isolated skyrmions are found in the same thickness regions. Due to the reduced TC of the 35 nm region, similar behaviour was observed in the 175 K and 180 K datasets, in Fig. 3h, n.

Lorentz transmission electron microscopy (LTEM) of a comparable FGT flake was performed to investigate the character of the domain walls (see Supplementary Note 5). Figure 3u–x shows the walls becoming more widely spaced and then disappearing as the applied magnetic field is increased following the ZFC procedure at 92 K. The change in image contrast by tilting by an angle α, shown in Fig. 3x, y indicates that our FGT samples exhibit Néel-type domain walls rather than Bloch-type (see Supplementary Figs. S11, S12), as observed in previous studies of thin flake samples33,35.

The real-space measurements also reveal that the average size of the stripe domains and skyrmions is increased at lower temperatures52. This result is quantified in Fig. 4a, where the average stripe domain size at 0 mT is plotted as a function of temperature (see Supplementary Note 6, Supplementary Figs. S13, S14). In addition, there is an indication that, for a fixed temperature, the thinner regions exhibit slightly larger domain sizes. The inset shows that a similar trend is seen for the average skyrmion diameter. The average domain wall width at 0 mT was also extracted, and is plotted as a function of temperature in Fig. 4b, showing a clear increase with decreasing temperature.

These results have important implications concerning the origin of the SD and skyrmion states. The observed temperature and thickness dependence of the domain size could be explained by the interplay of the dipolar interaction with the temperature-dependent uniaxial anisotropy observed in the magnetometry measurements. On the other hand, since Bloch-type domain walls are typically the lower energy configuration in out-of-plane magnetised films53, the LTEM observation of Néel-type domain walls is a strong indication that there must be some DMI contribution present to twist the SD and skyrmion helicity.

### Skyrmion field stability

To investigate the stability of the skyrmions, we studied their field evolution in detail, with the results shown in Fig. 5a–i. First, the FGT flake was initialised following the FC protocol. As the sample passes below TC under an applied field, the interplay of the Zeeman and dipolar interactions results in the formation of a dense disordered array of skyrmions, which survives down to low temperatures beyond their formation region, as shown for a FC field of 15 mT in Fig. 5b. We utilised image recognition software to extract the number and diameter d of the skyrmions in each image (see Supplementary Note 7, Supplementary Figs. S15, S16). These quantities have a strong dependence on the cooling field, as shown by the plots in Fig. 5f and g. For the 35 nm region, skyrmions were observed only when applying cooling fields of 15 mT and below.

After the 15 mT FC initialisation, images were acquired for both increasing and decreasing applied field, with the results shown in Fig. 5a–e. The number and d of the skyrmions in each thickness region during the field sweep are plotted in 5h and i. For increasing fields, the number of skyrmions quickly decreases, while the average size of the skyrmions slightly decreases. For decreasing field, the number of skyrmions also decreases, while d strongly increases. The filled coloured regions indicate the minimum and maximum diameter of the observed skyrmions at each field, showing that the increasing average size is largely driven by an expansion of a subset of the skyrmions, while the smallest skyrmions remain at a similar d, perhaps due to pinning effects.

In their theoretical treatment of skyrmion formation, Büttner et al. laid out useful indicators to distinguish between DMI and dipolar stabilised skyrmions54. The model predicts that DMI-stabilised skyrmions should not significantly alter their size as a function of the applied magnetic field, whereas skyrmions predominantly stabilised by the dipolar interaction can be expected to be sensitive to such field changes. Therefore, the present observations provide a strong indication that the dipolar interaction plays a dominant role in the stabilisation of the observed skyrmion states in FGT.

### Comparison to magnetic simulations

To validate this conclusion, we attempted to compare the experimental data with the results of micromagnetic simulations, utilising realistic values for the micromagnetic parameters (see methods). To determine the approximate value of interfacial DMI, D, required to form Néel-type domain walls in the system, we simulated skyrmion states with a range of D values and observed the helicity of the resulting magnetic textures. As shown in Fig. 6a–d, we found that, for our chosen micromagnetic parameters approximating the FGT flake, a value of D = 0.12 mJ/m2 was sufficient to alter the domain wall helicity from the expected Bloch-type to the experimentally observed Néel-type. The particular value will depend on factors such as the values of the other micromagnetic parameters, and the sample thickness. Selecting this parameter, we modelled the experimental FC results observed in Fig. 5a–e by nucleating a disordered array of skyrmions at 20 mT, and simulated field sweeps with both increasing and decreasing field, as shown in Fig. 6e. Similarly to the experiment, the skyrmions shrink for increasing field, and grow and form patch-like structures for negative magnetic fields—typical signs of a predominantly dipolar-stabilised spin texture54.

Since it is challenging to model temperature with micromagnetic simulations, we performed complementary mean-field calculations to study the temperature-dependent domain evolution observed in the field-sweep measurements in Fig. 3 (see methods). Optimal mean-field model parameters were tuned such that the field sweeps produced domain structures comparable to the experiment, and were estimated to be consistent with micromagnetic values (see “Methods”, Supplementary Note 8). In order to recreate the experimentally observed temperature dependence of the SD and skyrmion formation, it was necessary to scale the uniaxial anisotropy, K, with respect to the saturation magnetisation. Utilising a classical Callen-Callen power law scaling of K, the model captured the high-temperature behaviour well, but did not show the transition to monodomain switching at low temperature (see Supplementary Fig. S17). However, inspired by the large increase in uniaxial anisotropy at low temperatures observed in the magnetometry measurements, we included an additional temperature-dependent prefactor to the scaling law to further increase the anisotropy at lower temperatures (See Supplementary Note 9 and Supplementary Fig. S18). Together with an increase in the exchange interaction at lower temperatures, the model qualitatively reproduces the experimental FS phase diagram, including additional details such as the observed increase in domain size and crossover to monodomain behaviour at lower temperatures, as shown in Fig. 6g–i.

## Discussion

The combination of real-space imaging and simulation methods allows us to clarify the origin of skyrmion formation in FGT flakes. Specifically, the observed temperature dependence of the typical stripe domain size, and the field dependence of the skyrmion diameters, indicates that the spin textures are primarily stabilised by the dipolar interaction. The mean-field simulations support this conclusion, demonstrating that the temperature-dependent formation of skyrmions and SDs observed in the FGT flakes can be well-described by the large increase in the out-of-plane anisotropy with decreasing temperature. However, the LTEM observation of Néel-type domain walls necessitates the presence of some form of DMI contribution, although this is not required to be large. The micromagnetic simulations estimate a lower bound for the strength of the DMI in our sample, with the value of 0.12 mJ/m2 being much lower than the theoretical maximum value from previous DFT calculations of oxidised FGT, which suggested values could be as high as 2 mJ/m2 or more33. While the value we present is only a lower bound estimation, our results show that the history-dependent behaviour of FGT can be largely explained by considering the dominant dipolar and anisotropy effects. In the future, investigating a wider range of thicknesses, including down to the monolayer, may yield deeper insights into the spin texture formation, and reveal whether skyrmions can be stabilised at the single-layer limit.

We note that observations of Bloch-type skyrmions in FGT have exclusively been in thicker (>150 nm) samples32,35. With this in mind, we tentatively suggest that in such thick samples, any present interfacial DMI from the oxide layer may no longer be sufficient to stabilise Néel-type domain walls, and this would provide strong indication that the DMI observed in FGT flakes must be due to some interfacial effect. However, recent studies have suggested that the DMI could instead be due to some intrinsic property of the bulk material, either due to vacancy driven DMI38 or additional higher order effects39. Further work may be required to elucidate the true origin of the monochrial Néel-type domain walls. The issue of distinguishing between dipolar-stabilised and DMI-stabilised skyrmions can be considered controversial because it has been known for decades that the dipolar interaction and a strong uniaxial anisotropy can stabilise magnetic bubbles55. Due to their topological equivalence, these objects have thus been referred to as skyrmion bubbles56. However, the identification of dipolar interaction-stabilised skyrmions need not imply inferiority, since their discovery in new materials, particularly in 2D magnets, offers both renewed fundamental interest and fresh possibilities for their technological implementation. Further work will uncover whether similar history-dependent behaviour may be observed in other skyrmion-hosting 2D magnets. Moreover, the future discovery and investigation of new 2D magnetic materials, such as those proposed to possess chiral crystal structures57, or 2D multiferroics exhibiting electric field-tunable DMI58,59, may allow for the observation of further novel topological spin textures in 2D materials.

In summary, we utilised magnetic imaging techniques to explore the formation of magnetic stripe domains and skyrmions in an exfoliated FGT flake. We determined thickness-dependent magnetic phase diagrams following three distinct measurement protocols, revealing significant differences to other skyrmion hosts, such as bulk chiral and multilayer systems, evidenced by the possibility to selectively stabilise the stripe domain, skyrmion and uniform states at the same temperature and field point. Comparison of the real-spaces images and simulations suggests that the interplay of the dipolar interaction with the temperature-dependent uniaxial anisotropy is the primary stabilisation mechanism of the observed spin textures, while a DMI contribution must be present to twist the chirality of the domain walls from Bloch-type to the observed Néel-type. These results demonstrate the possibility to alter the properties of skyrmions in FGT by exploiting both DMI and dipolar stabilisation mechanisms, which can be readily controlled in 2D magnets thanks to their advantageous stacking properties34,36, opening the path towards advanced skyrmion-based spintronic devices composed of 2D magnet heterostructures.

## Methods

### Sample preparation and characterisation

The Fe3GeTe2 (FGT) single crystal was sourced from a commercial company, HQ Graphene. EDX measurements were performed with a Zeiss SEM Gemini-500, equipped with a Bruker XFlash 6-60 detector. Magnetometry measurements were carried out with a Quantum Design MPMS3 vibrating sample magnetometer, where the bulk FGT crystal was aligned along the relevant crystal axis and fixed to a quartz glass rod using GE varnish. Temperature and applied magnetic field were controlled by the built-in helium cryostat. The FGT flake sample for the STXM experiments was prepared by an all-dry viscoelastic transfer method. In the first step, the FGT bulk crystal was mechanically cleaved and exfoliated onto a PDMS stamp. A stepped flake with a thickness range between 20 and 200 nm, as estimated from the optical contrast, was selected and stamped onto a 100 nm thick SiN membrane. The flake was then capped by an exfoliated hexagonal boron nitride (hBN) sheet with a thickness of ~15 nm. The whole process was performed under ambient conditions, with each side of the flake being exposed to the atmosphere for approximately 30 min. Using a Bruker Dimension Icon atomic force microscope, measurements were performed on the capped FGT flake to determine the thickness of each region. Since the hBN capping complicates the thickness evaluation of the FGT sheet, the FGT thickness was estimated by averaging over 2–5 μm long steps. The thicknesses of all flake regions were then calibrated using measurements of the x-ray transmission (see Supplementary Note 2). For the preparation of the cross-sectional transmission electron microscope (TEM) sample, a focused ion beam (FIB) instrument (FEI FIB Scios) was used, operated at 30 kV for the initial cuts and subsequently reduced to 2 kV for thinning and cleaning. The transmission electron microscopy (TEM) investigation was conducted with a JEM-ARM200F (JEOL) microscope, operated at 200 kV and equipped with a cold field emission gun and a Cs-image aberration corrector.

### Total electron yield x-ray absorption spectroscopy

X-ray absorption spectroscopy measurements were performed on bulk FGT single crystals using the WERA beamline, at the KARA synchrotron in Karlsruhe Institute of Technology, which has an energy resolution of ΔE/E = 2 × 10−4. We utilised a superconducting magnet end station providing ultra-fast field switching with field ramping rates of 1.5 T/s. The FGT samples were glued to a conductive Mo sample holder, and electrical contact was ensured by coating the edges of the sample in carbon paste. A steel rod was then glued to the top of one of the crystal samples to enable in-vacuum cleaving. The samples were mounted inside the instrument, and the steel rod was removed, cleaving the FGT crystal sample within the vacuum chamber. Cooling was achieved with a liquid nitrogen cryostat. Absorption spectra were then measured over the oxygen K edge and the Fe L3 and L2 edges with fixed x-ray helicity. The energy was varied by controlling the monochromator at a speed of 0.2 eV/s to ensure no energy broadening. Meanwhile, the total electron yield (TEY) from the sample and incident x-ray intensity (I0 value) were measured using a Keithley 6517A electrometer. No noticeable energy drifts were observed between consecutive spectra.

### Scanning transmission x-ray microscopy

Scanning transmission x-ray microscopy (STXM) measurements were performed with the MAXYMUS instrument on the UE46 beamline at the BESSY II electron storage ring operated by the Helmholtz-Zentrum Berlin für Materialien und Energie. With the sample mounted inside the microscope, cooling was achieved by a He cryostat and the applied magnetic field was controlled by varying the arrangement of four permanent magnets. The x-ray beam was focused to a 20 nm spot size using a Fresnel zone plate and order selection aperture, setting the approximate spatial resolution. This focused beam, with a nominal x-ray energy of 707.5 eV, was then rastered across the sample pixel by pixel using a piezoelectric motor stage. By exploiting the effects of XMCD at the resonant x-ray energy at the Fe L3 edge, the transmission of the sample at each point was measured to form an image of the non-magnetic and magnetic domain structure, with the magnetic signal proportional to the out-of-plane magnetisation mz. The presented images of the magnetic domain structure were recorded using a single circular x-ray polarisation. Photons were counted by an avalanche photodiode.

### Lorentz electron transmission microscopy

LTEM measurements were performed using an FEI Titan3 transmission electron microscope equipped with a field-emission electron gun and operated at an acceleration voltage of 300 kV. In normal operation, the electromagnetic objective lens applies a 2 T field to the specimen which would force it into the saturated state. Instead, images were acquired in low-magnification mode where the image is formed using the diffraction lens and the objective lens weakly excited to apply a small magnetic field parallel to the electron beam. The applied field corresponding to a given objective lens current was calibrated to within 1 mT using a Thermo Fisher Hall probe holder. The specimen was cooled in-situ using a liquid nitrogen cooled Gatan double-tilt specimen holder which has a base temperature of 90 K. The FGT sample investigated by LTEM exhibited thickness regions between 15 and 70 nm.

Images were energy-filtered using a Gatan 865 Tridiem so that only electrons which had lost between 0 and 10 eV upon passing through the specimen contributed to the image, and an aperture subtending a half-angle of 0.14 mrad was used to ensure that only the 000 beam and the beams associated with magnetic scattering contributed to the image. Images were recorded on a 2048 by 2048 pixel charge-coupled device (CCD). The defocus and magnification were calibrated by acquiring images with the same lens settings from Agar Scientific’s S106 calibration specimen which consists of lines spaced by 463 nm ruled on an amorphous film. The defocus was found by taking digital Fourier transforms of these images and measuring the radii of the dark rings that result from the contrast transfer function of the lens60.

### Micromagnetic simulations

Micromagnetic simulations following the Landau-Lifschitz-Gilbert equation were performed using the MicroMagnum framework with custom extensions for the DMI. The simulations are based on the experimentally measured Ms = 250 kA/m, and utilised the real-space images of stripe domains to estimate the uniaxial anisotropy K = 44.2 kJ/m3 for the thin FGT flake and the exchange interaction A = 0.7 pJ/m. The anisotropy determined by SQUID measurements for a bulk crystal appears to significantly overestimate the value for a thin exfoliated flake, perhaps due to the enhanced shape anisotropy, which made this adjustment necessary. The system was simulated in two configurations, both with a cell size of 2 × 2 × 20 nm3. For Fig. 6a–d, the disk geometry was simulated by 50 × 50 × 1 cells, with only cells within a radius of 25 cells active. For the DMI estimation, varying DMI values were tested in 0.01 mJ/m2 steps until the Neél configuration was realised at D = 0.12 mJ/m2. For Fig. 6e, the system, with dimensions of 150 × 150 × 1 cells, was initialised with a disordered array of skyrmions that were relaxed into their equilibrium state at 20 mT. From this initial state, the field was both increased and decreased, with the system relaxed at each field point.

### Mean-field simulations

To perform temperature-dependent simulations we developed a mean-field model based on the standard classical spin Hamiltonian61 with spins distributed on a two-dimensional hexagonal lattice of 30 × 30 spins (approx. 150 nm × 150 nm) with periodic boundary conditions. The mean-field energy reads:

$${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{MF}}}}}}}}}= -\frac{1}{2}{J}_{{{{{{{{\rm{E}}}}}}}}}(T)\mathop{\sum}\limits_{\langle ij\rangle }{{{{{{{{\bf{m}}}}}}}}}_{i}\cdot {{{{{{{{\bf{m}}}}}}}}}_{j}-\frac{1}{2}{J}_{{{{{{{{\rm{D}}}}}}}}}(T)\mathop{\sum}\limits_{\langle ij\rangle }{{{{{{{{\bf{d}}}}}}}}}_{ij}\cdot ({{{{{{{{\bf{m}}}}}}}}}_{i}\times {{{{{{{{\bf{m}}}}}}}}}_{j})-{J}_{{{{{{{{\rm{K}}}}}}}}}(T)\mathop{\sum}\limits_{i}{(\hat{{{{{{{{\bf{n}}}}}}}}}\cdot {\hat{{{{{{{{\bf{m}}}}}}}}}}_{i})}^{2}\\ -\mathop{\sum}\limits_{i}{{{{{{{{\bf{m}}}}}}}}}_{i}\cdot {{{{{{{\bf{B}}}}}}}}-\frac{1}{2}{J}_{{{{{{{{\rm{DP}}}}}}}}}\mathop{\sum}\limits_{ij}\left(-\frac{{{{{{{{{\bf{m}}}}}}}}}_{i}\cdot {{{{{{{{\bf{m}}}}}}}}}_{j}}{{r}_{ij}^{3}}+3\frac{({{{{{{{{\bf{m}}}}}}}}}_{i}\cdot {\hat{{{{{{{{\bf{r}}}}}}}}}}_{ij})({{{{{{{{\bf{m}}}}}}}}}_{j}\cdot {\hat{{{{{{{{\bf{r}}}}}}}}}}_{ij})}{{r}_{ij}^{3}}\right)$$
(1)

where the symbols mi represent mean-field spins, and the individual energy terms starting from the left correspond to exchange, DMI, uniaxial anisotropy, Zeeman energy, and finally the dipolar energy (see Supplementary Note 8). It is worth pointing out that by taking the small angle approximation and zero temperature limit, one reduces the above model to the standard micromagnetic energy. Supplementary Note 8 identifies the relationship between the model constants JE(T = 0), JD(T = 0), JK(T = 0) (in the units of Joule J) and their micromagnetic equivalents A (J/m), D (J/m2) and K (J/m3). The exchange and anisotropy energy constants JE(T) and JK(T) are temperature-dependent, scaled following a modified Callen-Callen power law (see Supplementary Note 9). The temperature-dependence of JD(T) is weak and will be ignored. Thus the DMI energy term depends on temperature only through moments mi. The DMI vector dij is oriented in the lattice plane and perpendicular to the line connecting two neighbouring spins i and j. The anisotropy unit vector $$\hat{{{{{{{{\bf{n}}}}}}}}}$$ is oriented along the $$\hat{{{{{{{{\bf{z}}}}}}}}}$$-axis perpendicular to the lattice plane. The symbols enclosed in angled brackets in the exchange and DMI energy terms imply summation over the nearest neighbour spins. The spin moment mi has temperature dependent magnitude normalised to vary between mi = ±1 according to the expression:

$${{{{{{{{\bf{m}}}}}}}}}_{i}={{{{{{{\mathcal{L}}}}}}}}\left(\beta | {{{{{{{{\bf{B}}}}}}}}}_{i}^{{{{{{{{\rm{e}}}}}}}}}| \right){\hat{{{{{{{{\bf{B}}}}}}}}}}_{i}^{{{{{{{{\rm{e}}}}}}}}}$$
(2)

where $${{{{{{{\mathcal{L}}}}}}}}(x)=\coth x-{x}^{-1}$$ is the Langevin function, $$\beta ={({k}_{{{{{{{{\rm{B}}}}}}}}}T)}^{-1}$$, kB is the Boltzmann constant, and T the temperature. The vector $${{{{{{{{\bf{B}}}}}}}}}_{i}^{{{{{{{{\rm{e}}}}}}}}}$$ represents the effective field acting on the spin mi and $$| {{{{{{{{\bf{B}}}}}}}}}_{i}^{{{{{{{{\rm{e}}}}}}}}}|$$ is the magnitude of $${{{{{{{{\bf{B}}}}}}}}}_{i}^{{{{{{{{\rm{e}}}}}}}}}$$. The expression for the effective field can be obtained from Eq. (1) by calculating the variational derivative $${{{{{{\bf{B}}}}}}}_{i}^{{{{{{\rm{e}}}}}}}=-\delta {{{{{{\mathcal{H}}}}}}}_{{{{{{\rm{MF}}}}}}}/\delta {{{{{{\bf{m}}}}}}}_{i}={J}_{{{{{{\rm{E}}}}}}}(T){\sum }_{j}{{{{{{\bf{m}}}}}}}_{j}-{J}_{{{{{{\rm{D}}}}}}}{\sum }_{j}{{{{{{\bf{d}}}}}}}_{ij}\times {{{{{{\bf{m}}}}}}}_{j}+2{J}_{{{{{{\rm{K}}}}}}}(T)(\hat{{{{{{\bf{n}}}}}}}\cdot {\hat{{{{{{\bf{m}}}}}}}}_{i})\hat{{{{{{\bf{n}}}}}}}+{{{{{\bf{B}}}}}}+{J}_{{{{{{\rm{DP}}}}}}}{\sum }_{j}({{{{{{\bf{m}}}}}}}_{j}+3({{{{{{\bf{m}}}}}}}_{j}\cdot {\hat{{{{{{\bf{r}}}}}}}}_{ij}){\hat{{{{{{\bf{r}}}}}}}}_{ij}){r}_{ij}^{-3}$$. Thus according to Eq. (2), the mean-field spin mi depends on the exchange and DMI energy couplings, material anisotropy, external and dipolar field, and also on the temperature T. The magnetisation structures for a given set of parameters are evaluated by minimising Eq. (1) during the field or temperature evolution starting from uniquely defined initial states61. The parameters required to generate the results in Fig. 6 were JE(0) = 0.7, JD = 0.1, JK(0) = 2.5, JDP = 0.35. It was also necessary to identify appropriate temperature dependencies of JE(T) and JK(T) (see Supplementary Note 9).


text:
How to solve this recurrence relation? (convolution integral)

Following is a recurrence relation written by Robert Israel from Poisson arrivals followed by locking

$u_n(t) = \int_0^{t-nT} \lambda \exp(-\lambda y) u_{n-1}(t-y-T)\, dy$.

The solution he wrote is:

$u_n(t) = 1 - q_n(t-nT) \exp(-\lambda (t-nT))$ where $q_n(s) = \sum_{j=0}^{n-1} (\lambda s)^j/j!$.

I wonder how the recurrence is solved and what kind of method it is?

The integral is a convolution. You can solve it using Laplace transforms (convolutions in the time domain correspond to products in the Laplace domain). The boundary condition $u_0(t)$ can be derived separately. If it helps consider $v_n(t) \equiv u_n(t+nT)$ – Emre Apr 25 '11 at 6:43
Following Emre's hint I write $v_n(t):=u_n(t+nT)$ and arbitrarily assume $v_0(t)\equiv 1$. The $v_n$ satisfy the simpler looking recursion $$v_n(t)=\int_0^t \lambda e^{-\lambda y} v_{n-1}(t-y) dy\ .\qquad(*)$$ Computing the first few $v_n$'s by hand one gets the idea that the $v_n$ might be of the form $v_n(t)=1-q_n(t)e^{-\lambda t}$ with polynomials $q_n(t)$. Entering this "Ansatz" into the recursion $(*)$ one obtains after some calculation the following recursion for the $q_n$: $$q_n(t)=1+\lambda \int_0^t q_{n-1}(y) dy\qquad (n\geq1)$$ with $q_0(t)\equiv0$. Now it is easy to see that the $q_n(t$) are the partial sums $s_{n-1}$ of the series for $e^{\lambda t}$.