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the matrix cookbook kaare brandt petersen michael syskind pedersen version february 16 2006 what is this these pages are a collection of facts identities approxima tions inequalities relations about matrices ...

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                                           The Matrix Cookbook
                                                   Kaare Brandt Petersen
                                                 Michael Syskind Pedersen
                                              Version: February 16, 2006
                            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 at cookbook@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 contribu-
                            tions and suggestions: Christian Rishøj, Douglas L. Theobald, Esben Hoegh-
                            Rasmussen, Lars Christiansen, and Vasile Sima. We would also like thank The
                            Oticon Foundation for funding our PhD studies.
                                                              1
                                   CONTENTS                                                                  CONTENTS
                                   Contents
                                   1 Basics                                                                               5
                                       1.1   Trace and Determinants . . . . . . . . . . . . . . . . . . . . . . .         5
                                       1.2   The Special Case 2x2 . . . . . . . . . . . . . . . . . . . . . . . . .       5
                                   2 Derivatives                                                                          7
                                       2.1   Derivatives of a Determinant . . . . . . . . . . . . . . . . . . . .         7
                                       2.2   Derivatives of an Inverse . . . . . . . . . . . . . . . . . . . . . . .      8
                                       2.3   Derivatives of Matrices, Vectors and Scalar Forms         . . . . . . . .    9
                                       2.4   Derivatives of Traces . . . . . . . . . . . . . . . . . . . . . . . . .     11
                                       2.5   Derivatives of Structured Matrices . . . . . . . . . . . . . . . . .        12
                                   3 Inverses                                                                           15
                                       3.1   Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   15
                                       3.2   Exact Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . .     16
                                       3.3   Implication on Inverses . . . . . . . . . . . . . . . . . . . . . . . .     17
                                       3.4   Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . .      17
                                       3.5   Generalized Inverse . . . . . . . . . . . . . . . . . . . . . . . . . .     17
                                       3.6   Pseudo Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . .      17
                                   4 Complex Matrices                                                                   19
                                       4.1   Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . .       19
                                   5 Decompositions                                                                     22
                                       5.1   Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . .        22
                                       5.2   Singular Value Decomposition . . . . . . . . . . . . . . . . . . . .        22
                                       5.3   Triangular Decomposition . . . . . . . . . . . . . . . . . . . . . .        24
                                   6 Statistics and Probability                                                         25
                                       6.1   Definition of Moments . . . . . . . . . . . . . . . . . . . . . . . .        25
                                       6.2   Expectation of Linear Combinations . . . . . . . . . . . . . . . .          26
                                       6.3   Weighted Scalar Variable      . . . . . . . . . . . . . . . . . . . . . .   27
                                   7 Gaussians                                                                          28
                                       7.1   Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    28
                                       7.2   Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       30
                                       7.3   Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     32
                                       7.4   Mixture of Gaussians . . . . . . . . . . . . . . . . . . . . . . . . .      33
                                   8 Special Matrices                                                                   34
                                       8.1   Units, Permutation and Shift . . . . . . . . . . . . . . . . . . . .        34
                                       8.2   The Singleentry Matrix      . . . . . . . . . . . . . . . . . . . . . . .   35
                                       8.3   Symmetric and Antisymmetric . . . . . . . . . . . . . . . . . . .           37
                                       8.4   Vandermonde Matrices . . . . . . . . . . . . . . . . . . . . . . . .        37
                                       8.5   Toeplitz Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . .     38
                                       8.6   The DFT Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .        39
                                     Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 2
                                   CONTENTS                                                                   CONTENTS
                                       8.7   Positive Definite and Semi-definite Matrices . . . . . . . . . . . .           40
                                       8.8   Block matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . .       41
                                   9 Functions and Operators                                                              43
                                       9.1   Functions and Series . . . . . . . . . . . . . . . . . . . . . . . . .       43
                                       9.2   Kronecker and Vec Operator         . . . . . . . . . . . . . . . . . . . .   44
                                       9.3   Solutions to Systems of Equations        . . . . . . . . . . . . . . . . .   45
                                       9.4   Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       47
                                       9.5   Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     48
                                       9.6   Integral Involving Dirac Delta Functions . . . . . . . . . . . . . .         48
                                       9.7   Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      49
                                   A One-dimensional Results                                                              50
                                       A.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       50
                                       A.2 One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . .             51
                                   B Proofs and Details                                                                   53
                                       B.1 Misc Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        53
                                     Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 3
                               CONTENTS                                                        CONTENTS
                               Notation and Nomenclature
                                    A       Matrix
                                   A        Matrix indexed for some purpose
                                     ij
                                    A       Matrix indexed for some purpose
                                      i
                                     ij
                                   A        Matrix indexed for some purpose
                                   An       Matrix indexed for some purpose or
                                            The n.th power of a square matrix
                                     −1
                                   A        The inverse matrix of the matrix A
                                   A+       The pseudo inverse matrix of the matrix A (see Sec. 3.6)
                                   A1/2     The square root of a matrix (if unique), not elementwise
                                  (A)       The (i,j).th entry of the matrix A
                                      ij
                                   Aij      The (i,j).th entry of the matrix A
                                   [A]      The ij-submatrix, i.e. A with i.th row and j.th column deleted
                                      ij
                                    a       Vector
                                    ai      Vector indexed for some purpose
                                    ai      The i.th element of the vector a
                                    a       Scalar
                                    ℜz      Real part of a scalar
                                    ℜz      Real part of a vector
                                   ℜZ       Real part of a matrix
                                    ℑz      Imaginary part of a scalar
                                    ℑz      Imaginary part of a vector
                                   ℑZ       Imaginary part of a matrix
                                  det(A)    Determinant of A
                                  Tr(A)     Trace of the matrix A
                                 diag(A)    Diagonal matrix of the matrix A, i.e. (diag(A))  =δ A
                                                                                           ij   ij  ij
                                  vec(A)    The vector-version of the matrix A (see Sec. 9.2.2)
                                   ||A||    Matrix norm (subscript if any denotes what norm)
                                     T
                                   A        Transposed matrix
                                     ∗
                                    A       Complex conjugated matrix
                                     H
                                   A        Transposed and complex conjugated matrix (Hermitian)
                                  A◦B Hadamard(elementwise) product
                                  A⊗B Kroneckerproduct
                                    0       The null matrix. Zero in all entries.
                                     I      The identity matrix
                                     ij
                                    J       The single-entry matrix, 1 at (i,j) and zero elsewhere
                                    Σ       Apositive definite matrix
                                    Λ       Adiagonal matrix
                                Petersen & Pedersen, The Matrix Cookbook, Version: February 16, 2006, Page 4
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...The matrix cookbook kaare brandt petersen michael syskind pedersen version february 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 keywords algebra derivative determinant inverse dierentiate acknowledgements like thank following contribu christian rishoj douglas l theobald esben hoegh rasmussen lars christiansen vasile sima als...

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