# The Matrix Cookbook

## Chapter 2

22 mai 2017 be seen from the second derivative (if it exists). 6. Page 7. DMM summer 2017 ... Calculate the gradient of fA
optimization

## The Matrix Cookbook

15 nov. 2012 determinant derivative of inverse matrix
matrixcookbook

b − Ax2. 2. = (b − Ax)T (b − Ax). = bT b − (Ax)T b − bT Ax + xT AT Ax. = bT b − 2bT Ax + xT AT Ax Because mixed second partial derivatives satisfy.
lecture

## The Matrix Cookbook

determinant derivative of inverse matrix
Matrix Cookbook

## Techniques of Integration

apparent that the function you wish to integrate is a derivative in some EXAMPLE 8.1.2 Evaluate ∫ sin(ax + b) dx assuming that a and b are constants ...
calculus Techniques of Integration

## Week 3 Quiz: Differential Calculus: The Derivative and Rules of

Question 2: Find limx→2f(x): f(x) = 1776. (A) +∞. (B) 1770. (C) −∞. (D) Does not exist! (E) None of the above. Answer: (E) The limit of any constant

## Assignment 2 — Solutions

If a1b2 = a2b1 show that this equation reduces to the form y′ = g(ax + by). Solution Substituting a2 = λa1 and b2 = λb1 into the equation yields:.
Assignment Solutions

## Introduction to Linear Algebra 5th Edition

To see that action I will write b1
linearalgebra

1 mar. 2016
ORF S Lec gh

## Order and Degree and Formation of Partial Differential Equations

When a differential equation contains one or more partial derivatives of an (viii) z = ax e' +. 1. 2. Sol. (1) We are given z = (2x + a) (2 y + b).
partial differential equations unit

146106

### The Matrix Cookbook

[ http://matrixcookbook.com ]

### Version: September 5, 2007

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:The identities, 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 contributions and suggestions: Bill Baxter, Christian Rishøj, Douglas L. Theobald, Esben Hoegh-Rasmussen, Jan Larsen, Korbinian Strimmer, Lars Christiansen, Lars Kai Hansen, Leland Wilkinson, Liguo He, Loic Thibaut, Ole Winther, Stephan Hattinger, and Vasile Sima. We would also like thank The Oticon Foundation for funding our PhD studies. 1

### A One-dimensional Results 59

A.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 A.2 One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . . 60

### B Proofs and Details 62

B.1 Misc Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 2007,Page 3

### AMatrix

A ijMatrix indexed for some purpose A iMatrix indexed for some purpose A ijMatrix indexed for some purpose A nMatrix indexed for some purposeor

### The n.th power of a square matrix

A -1The inverse matrix of the matrixA A +The pseudo inverse matrix of the matrixA(see Sec. 3.6) A

#### 1/2The square root of a matrix (if unique), not elementwise

(A)ijThe (i,j).th entry of the matrixA A ijThe (i,j).th entry of the matrixA [A]ijTheij-submatrix, i.e.Awith i.th row and j.th column deleted aVector a iVector indexed for some purpose a iThe i.th element of the vectora aScalar ?zReal part of a scalar ?zReal part of a vector ?ZReal part of a matrix ?zImaginary part of a scalar ?zImaginary part of a vector ?ZImaginary part of a matrix det(A) Determinant ofA

### Tr(A) Trace of the matrixA

diag(A) Diagonal matrix of the matrixA, i.e. (diag(A))ij=δijAij vec(A) The vector-version of the matrixA(see Sec. 10.2.2) sup Supremum of a set ||A||Matrix norm (subscript if any denotes what norm) A

### TTransposed matrix

A ?Complex conjugated matrix A HTransposed and complex conjugated matrix (Hermitian)

### The Matrix Cookbook

[ http://matrixcookbook.com ]

### Version: September 5, 2007

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:The identities, 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 contributions and suggestions: Bill Baxter, Christian Rishøj, Douglas L. Theobald, Esben Hoegh-Rasmussen, Jan Larsen, Korbinian Strimmer, Lars Christiansen, Lars Kai Hansen, Leland Wilkinson, Liguo He, Loic Thibaut, Ole Winther, Stephan Hattinger, and Vasile Sima. We would also like thank The Oticon Foundation for funding our PhD studies. 1

### A One-dimensional Results 59

A.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 A.2 One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . . 60

### B Proofs and Details 62

B.1 Misc Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 2007,Page 3

### AMatrix

A ijMatrix indexed for some purpose A iMatrix indexed for some purpose A ijMatrix indexed for some purpose A nMatrix indexed for some purposeor

### The n.th power of a square matrix

A -1The inverse matrix of the matrixA A +The pseudo inverse matrix of the matrixA(see Sec. 3.6) A

#### 1/2The square root of a matrix (if unique), not elementwise

(A)ijThe (i,j).th entry of the matrixA A ijThe (i,j).th entry of the matrixA [A]ijTheij-submatrix, i.e.Awith i.th row and j.th column deleted aVector a iVector indexed for some purpose a iThe i.th element of the vectora aScalar ?zReal part of a scalar ?zReal part of a vector ?ZReal part of a matrix ?zImaginary part of a scalar ?zImaginary part of a vector ?ZImaginary part of a matrix det(A) Determinant ofA

### Tr(A) Trace of the matrixA

diag(A) Diagonal matrix of the matrixA, i.e. (diag(A))ij=δijAij vec(A) The vector-version of the matrixA(see Sec. 10.2.2) sup Supremum of a set ||A||Matrix norm (subscript if any denotes what norm) A

### TTransposed matrix

A ?Complex conjugated matrix A HTransposed and complex conjugated matrix (Hermitian)