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by Kaare B Petersen & Michael S Pedersen

TOC

1 Basics
1.1 Trace and Determinants
1.2 The Special Case 2×2

2 Derivatives
2.1 Derivatives of a Determinant
2.2 Derivatives of an Inverse
2.3 Derivatives of Eigenvalues
2.4 Derivatives of Matrices, Vectors and Scalar Forms
2.5 Derivatives of Traces
2.6 Derivatives of vector norms
2.7 Derivatives of matrix norms
2.8 Derivatives of Structured Matrices

3 Inverses
3.1 Basic
3.2 Exact Relations
3.3 Implication on Inverses
3.4 Approximations
3.5 Generalized Inverse
3.6 Pseudo Inverse

4 Complex Matrices
4.1 Complex Derivatives
4.2 Higher order and non-linear derivatives
4.3 Inverse of complex sum

5 Solutions and Decompositions
5.1 Solutions to linear equations
5.2 Eigenvalues and Eigenvectors
5.3 Singular Value Decomposition
5.4 Triangular Decomposition
5.5 LU decomposition
5.6 LDM decomposition
5.7 LDL decompositions

6 Statistics and Probability
6.1 Definition of Moments
6.2 Expectation of Linear Combinations
6.3 Weighted Scalar Variable

7 Multivariate Distributions
7.1 Cauchy
7.2 Dirichlet
7.3 Normal
7.4 Normal-Inverse Gamma
7.5 Gaussian
7.6 Multinomial
7.7 Student’s t
7.8 Wishart
7.9 Wishart, Inverse8

8 Gaussians
8.1 Basics
8.2 Moments
8.3 Miscellaneous
8.4 Mixture of Gaussians

9 Special Matrices
9.1 Block matrices
9.2 Discrete Fourier Transform Matrix, The
9.3 Hermitian Matrices and skew-Hermitian
9.4 Idempotent Matrices
9.5 Orthogonal matrices
9.6 Positive Definite and Semi-definite Matrices
9.7 Singleentry Matrix, The
9.8 Symmetric, Skew-symmetric/Antisymmetric
9.9 Toeplitz Matrices
9.10 Transition matrices
9.11 Units, Permutation and Shift
9.12 Vandermonde Matrices

10 Functions and Operators
10.1 Functions and Series
10.2 Kronecker and Vec Operator
10.3 Vector Norms
10.4 Matrix Norms
10.5 Rank
10.6 Integral Involving Dirac Delta Functions
10.7 Miscellaneous

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