# 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

## Gradients of Inner Products

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
week answers

## 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

271550

### The Matrix Cookbook

[ http://matrixcookbook.com ]

1

### Introduction

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 acookbook@2302.dk. Keywords:Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, dierentiate a matrix. Acknowledgements:We would like to thank the following for contributions and suggestions: Bill Baxter, Brian Templeton, Christian Rishj, Christian Schroppel, Dan Boley, Douglas L. Theobald, Esben Hoegh-Rasmussen, Evripidis Karseras, Georg Martius, Glynne Casteel, Jan Larsen, Jun Bin Gao, Jurgen Struckmeier, Kamil Dedecius, Karim T. Abou-Moustafa, Korbinian Strimmer, Lars Christiansen, Lars Kai Hansen, Leland Wilkinson, Liguo He, Loic Thibaut, Markus Froeb, Michael Hubatka, Miguel Bar~ao, Ole Winther, Pavel Sakov, Stephan Hattinger, Troels Pedersen, Vasile Sima, Vincent Rabaud, Zhaoshui He. We would also like thank The Oticon Foundation for funding our PhD studies.Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012,Page 2

### A One-dimensional Results 64

A.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 A.2 One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . . 65

### B Proofs and Details 66

B.1 Misc Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012,Page 4

### 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 (column-vector) a iVector indexed for some purpose

### The Matrix Cookbook

[ http://matrixcookbook.com ]

1

### Introduction

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 acookbook@2302.dk. Keywords:Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, dierentiate a matrix. Acknowledgements:We would like to thank the following for contributions and suggestions: Bill Baxter, Brian Templeton, Christian Rishj, Christian Schroppel, Dan Boley, Douglas L. Theobald, Esben Hoegh-Rasmussen, Evripidis Karseras, Georg Martius, Glynne Casteel, Jan Larsen, Jun Bin Gao, Jurgen Struckmeier, Kamil Dedecius, Karim T. Abou-Moustafa, Korbinian Strimmer, Lars Christiansen, Lars Kai Hansen, Leland Wilkinson, Liguo He, Loic Thibaut, Markus Froeb, Michael Hubatka, Miguel Bar~ao, Ole Winther, Pavel Sakov, Stephan Hattinger, Troels Pedersen, Vasile Sima, Vincent Rabaud, Zhaoshui He. We would also like thank The Oticon Foundation for funding our PhD studies.Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012,Page 2

### A One-dimensional Results 64

A.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 A.2 One Dimensional Mixture of Gaussians . . . . . . . . . . . . . . . 65

### B Proofs and Details 66

B.1 Misc Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Petersen & Pedersen, The Matrix Cookbook, Version: November 15, 2012,Page 4

### 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 (column-vector) a iVector indexed for some purpose