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the matrix cookbook kaare brandt petersen michael syskind pedersen version january 5 2005 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: January 5, 2005
                            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 kbp@imm.dtu.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 kbp@imm.dtu.dk.
                            Acknowledgements: We would like to thank the following for discussions,
                            proofreading, extensive corrections and suggestions: Esben Hoegh-Rasmussen
                            and Vasile Sima.
                            Keywords: Matrix algebra, matrix relations, matrix identities, derivative of
                            determinant, derivative of inverse matrix, di®erentiate a matrix.
                                                              1
                                  CONTENTS                                                                 CONTENTS
                                  Contents
                                  1 Basics                                                                              5
                                      1.1   Trace and Determinants . . . . . . . . . . . . . . . . . . . . . . .        5
                                      1.2   The Special Case 2x2 . . . . . . . . . . . . . . . . . . . . . . . . .      6
                                  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                                                                           14
                                      3.1   Exact Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . .    14
                                      3.2   Implication on Inverses . . . . . . . . . . . . . . . . . . . . . . . .    14
                                      3.3   Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . .     15
                                      3.4   Generalized Inverse . . . . . . . . . . . . . . . . . . . . . . . . . .    15
                                      3.5   Pseudo Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . .     15
                                  4 Complex Matrices                                                                   17
                                      4.1   Complex Derivatives . . . . . . . . . . . . . . . . . . . . . . . . .      17
                                  5 Decompositions                                                                     20
                                      5.1   Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . .       20
                                      5.2   Singular Value Decomposition . . . . . . . . . . . . . . . . . . . .       20
                                      5.3   Triangular Decomposition . . . . . . . . . . . . . . . . . . . . . .       21
                                  6 General Statistics and Probability                                                 22
                                      6.1   Moments of any distribution . . . . . . . . . . . . . . . . . . . . .      22
                                      6.2   Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     22
                                  7 Gaussians                                                                          24
                                      7.1   One Dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . .      24
                                      7.2   Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   25
                                      7.3   Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      27
                                      7.4   Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    29
                                      7.5   One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . .         29
                                      7.6   Mixture of Gaussians . . . . . . . . . . . . . . . . . . . . . . . . .     30
                                  8 Miscellaneous                                                                      31
                                      8.1   Functions and Series . . . . . . . . . . . . . . . . . . . . . . . . .     31
                                      8.2   Indices, Entries and Vectors . . . . . . . . . . . . . . . . . . . . .     32
                                      8.3   Solutions to Systems of Equations       . . . . . . . . . . . . . . . . .  35
                                      8.4   Block matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . .     36
                                      8.5   Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     37
                                      8.6   Positive De¯nite and Semi-de¯nite Matrices . . . . . . . . . . . .         38
                                      Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 2
                                  CONTENTS                                                                CONTENTS
                                      8.7  Integral Involving Dirac Delta Functions . . . . . . . . . . . . . .      39
                                      8.8  Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   40
                                  A Proofs and Details                                                               41
                                     Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 3
                           CONTENTS                                                 CONTENTS
                           Notation and Nomenclature
                                   A     Matrix
                                  A      Matrix indexed for some purpose
                                    ij
                                  Ai     Matrix indexed for some purpose
                                    ij
                                  A      Matrix indexed for some purpose
                                  An     Matrix indexed for some purpose or
                                         The n.th power of a square matrix
                                  A−1    The inverse matrix of the matrix A
                                    +
                                  A      The pseudo inverse matrix of the matrix A
                                  A1/2   The square root of a matrix (if unique), not elementwise
                                 (A)ij   The (i,j).th entry of the matrix A
                                  Aij    The (i,j).th entry of the matrix A
                                   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
                                  ||A||  Matrix norm (subscript if any denotes what norm)
                                  AT     Transposed matrix
                                  A∗     Complex conjugated matrix
                                  AH     Transposed and complex conjugated matrix
                                 A◦B Hadamard(elementwise) product
                                 A⊗B Kronecker product
                                   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
                                   §     Apositive de¯nite matrix
                                   ¤     Adiagonal matrix
                              Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 4
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...The matrix cookbook kaare brandt petersen michael syskind pedersen version january 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 kbp imm dtu dk its ongoing project keeping repository involving naturally will apparent date header suggestions your suggestion additional content or elaboration some topics most welcome acknowledgements like thank following discussions proofreading extensive esben hoegh rasmussen vasile sima keywords algebra derivative determinant inverse di erentiate contents basics tra...

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