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The Matrix Cookbook [ http://matrixcookbook.com ] Kaare Brandt Petersen Michael Syskind Pedersen Version: November 15, 2012 1 Introduction What is this? These pages are a collection of facts (identities, approxima- tions, inequalities, relations, ...) about matrices and matters relating to them. It is collected in this form for the convenience of anyone who wants a quick desktop reference . Disclaimer: Theidentities, approximations and relations presented here were obviously not invented but collected, borrowed and copied from a large amount of sources. These sources include similar but shorter notes found on the internet and appendices in books - see the references for a full list. Errors: Very likely there are errors, typos, and mistakes for which we apolo- gize and would be grateful to receive corrections at cookbook@2302.dk. Its ongoing: The project of keeping a large repository of relations involving matrices is naturally ongoing and the version will be apparent from the date in the header. Suggestions: Your suggestion for additional content or elaboration of some topics is most welcome acookbook@2302.dk. Keywords: Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, differentiate a matrix. Acknowledgements: We would like to thank the following for contributions and suggestions: Bill Baxter, Brian Templeton, Christian Rishøj, Christian Schr¨oppel, Dan Boley, Douglas L. Theobald, Esben Hoegh-Rasmussen, Evripidis Karseras, Georg Martius, Glynne Casteel, Jan Larsen, Jun Bin Gao, Jurgen¨ Struckmeier, Kamil Dedecius, Karim T. Abou-Moustafa, Korbinian Strimmer, Lars Christiansen, Lars Kai Hansen, Leland Wilkinson, Liguo He, Loic Thibaut, Markus Froeb, Michael Hubatka, Miguel Bar˜ao, Ole Winther, Pavel Sakov, Stephan Hattinger, Troels Pedersen, Vasile Sima, Vincent Rabaud, Zhaoshui He. We would also like thank The Oticon Foundation for funding our PhD studies. Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 2 CONTENTS CONTENTS Contents 1 Basics 6 1.1 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 The Special Case 2x2 . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Derivatives 8 2.1 Derivatives of a Determinant . . . . . . . . . . . . . . . . . . . . 8 2.2 Derivatives of an Inverse . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Derivatives of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Derivatives of Matrices, Vectors and Scalar Forms . . . . . . . . 10 2.5 Derivatives of Traces . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 Derivatives of vector norms . . . . . . . . . . . . . . . . . . . . . 14 2.7 Derivatives of matrix norms . . . . . . . . . . . . . . . . . . . . . 14 2.8 Derivatives of Structured Matrices . . . . . . . . . . . . . . . . . 14 3 Inverses 17 3.1 Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Exact Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Implication on Inverses . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 Generalized Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.6 Pseudo Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 Complex Matrices 24 4.1 Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 Higher order and non-linear derivatives . . . . . . . . . . . . . . . 26 4.3 Inverse of complex sum . . . . . . . . . . . . . . . . . . . . . . . 27 5 Solutions and Decompositions 28 5.1 Solutions to linear equations . . . . . . . . . . . . . . . . . . . . . 28 5.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . 30 5.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . 31 5.4 Triangular Decomposition . . . . . . . . . . . . . . . . . . . . . . 32 5.5 LUdecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.6 LDMdecomposition . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.7 LDLdecompositions . . . . . . . . . . . . . . . . . . . . . . . . . 33 6 Statistics and Probability 34 6.1 Definition of Moments . . . . . . . . . . . . . . . . . . . . . . . . 34 6.2 Expectation of Linear Combinations . . . . . . . . . . . . . . . . 35 6.3 Weighted Scalar Variable . . . . . . . . . . . . . . . . . . . . . . 36 7 Multivariate Distributions 37 7.1 Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 7.2 Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 7.3 Normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 7.4 Normal-Inverse Gamma . . . . . . . . . . . . . . . . . . . . . . . 37 7.5 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 7.6 Multinomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 3 CONTENTS CONTENTS 7.7 Student’s t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 7.8 Wishart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.9 Wishart, Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 8 Gaussians 40 8.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 8.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 8.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 8.4 Mixture of Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . 44 9 Special Matrices 46 9.1 Block matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 9.2 Discrete Fourier Transform Matrix, The . . . . . . . . . . . . . . 47 9.3 Hermitian Matrices and skew-Hermitian . . . . . . . . . . . . . . 48 9.4 Idempotent Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 49 9.5 Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . 49 9.6 Positive Definite and Semi-definite Matrices . . . . . . . . . . . . 50 9.7 Singleentry Matrix, The . . . . . . . . . . . . . . . . . . . . . . . 52 9.8 Symmetric, Skew-symmetric/Antisymmetric . . . . . . . . . . . . 54 9.9 Toeplitz Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 9.10 Transition matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 55 9.11 Units, Permutation and Shift . . . . . . . . . . . . . . . . . . . . 56 9.12 Vandermonde Matrices . . . . . . . . . . . . . . . . . . . . . . . . 57 10 Functions and Operators 58 10.1 Functions and Series . . . . . . . . . . . . . . . . . . . . . . . . . 58 10.2 Kronecker and Vec Operator . . . . . . . . . . . . . . . . . . . . 59 10.3 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.4 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.5 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 10.6 Integral Involving Dirac Delta Functions . . . . . . . . . . . . . . 62 10.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 A One-dimensional Results 64 A.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 A.2 One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . . 65 B Proofs and Details 66 B.1 Misc Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012, Page 4
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