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the matrix cookbook 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 ...

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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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...The matrix cookbook kaare brandt petersen michael syskind pedersen version november introduction what is this these pages are a collection of facts identities approxima tions inequalities relations about matrices and matters relating to them it collected in form for convenience anyone who wants quick desktop reference disclaimer theidentities approximations presented here were obviously not invented but borrowed copied from large amount sources include similar shorter notes found on internet appendices books see references full list errors very likely there typos mistakes which we apolo gize would be grateful receive corrections at dk its ongoing project keeping repository involving naturally will apparent date header suggestions your suggestion additional content or elaboration some topics most welcome acookbook keywords algebra derivative determinant inverse dierentiate acknowledgements like thank following contributions bill baxter brian templeton christian rishoj schr oppel dan bol...

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