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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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