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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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