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    Kronecker Products and Matrix Calculus with Applications
    Kronecker Products and Matrix Calculus with Applications
    Kronecker Products and Matrix Calculus with Applications
    Ebook201 pages1 hour

    Kronecker Products and Matrix Calculus with Applications

    By Alexander Graham

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    About this ebook

    Enhanced by many worked examples — as well as problems and solutions — this in-depth text discusses the Kronecker matrix product. Named after a 19th-century German mathematician, Leopold Kronecker, the Kronecker product is an increasingly important and useful matrix operation and an area of matrix calculus with numerous applications. A great deal of information previously only available in specialized and hard-to-find texts and papers is presented here at an undergraduate level.
    After the preliminaries of an introductory chapter, the treatment proceeds to examinations of the Kronecker product and its applications. An introduction to matrix calculus is followed by further development of the subject, including an application of Kronecker products. Final chapters explore the derivative of a matrix and some applications of matrix calculus. The text concludes with solutions to the problems that appear at the end of each chapter and helpful tables of formulae and derivatives.
    LanguageEnglish
    PublisherDover Publications
    Release dateJun 13, 2018
    ISBN9780486831206
    Kronecker Products and Matrix Calculus with Applications

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      Book preview

      Kronecker Products and Matrix Calculus with Applications - Alexander Graham

      KRONECKER PRODUCTS & MATRIX CALCULUS WITH APPLICATIONS

      KRONECKER PRODUCTS & MATRIX CALCULUS WITH APPLICATIONS

      ALEXANDER GRAHAM

      M.A., M.Sc., Ph.D.

      Dover Publications, Inc.

      Mineola, New York

      Copyright

      Copyright © 1981, 2018 by

      Alexander Graham All rights reserved.

      Bibliographical Note

      This Dover edition, first published in 2018, is a slightly corrected republication of the work originally published by Ellis Horwood Limited, Chichester, West Sussex, England, in 1981.

      International Standard Book Number

      ISBN-13: 978-0-486-82417-8

      ISBN-10: 0-486-82417-9

      Manufactured in the United States by LSC Communications

      82417901 2018

      www.doverpublications.com

      Table of Contents

      Author’s Preface

      Symbols and Notation Used

      Chapter 1 – Preliminaries

      1.1 Introduction

      1.2 Unit Vectors and Elementary Matrices

      1.3 Decompositions of a Matrix

      1.4 The Trace Function

      1.5 The Vec Operator

      Problems for Chapter 1

      Chapter 2 – The Kronecker Product

      2.1 Introduction

      2.2 Definition of the Kronecker Product

      2.3 Some Properties and Rules for Kronecker Products

      2.4 Definition of the Kronecker Sum

      2.5 The Permutation Matrix associating vec X and vec X ′

      Problems for Chapter 2

      Chapter 3 – Some Applications of the Kronecker Product

      3.1 Introduction

      3.2 The Derivative of a Matrix

      3.3 Problem 1: solution of AX + XB = C

      3.4 Problem 2: solution of AX − XA = μX

      3.7 Problem 5: solution of AXB = C

      3.8 Problem 6: Pole assignment for a Multivariable System

      Chapter 4 – Introduction to Matrix Calculus

      4.1 Introduction

      4.2 The Derivatives of Vectors

      4.3 The Chain rule for Vectors

      4.4 The Derivative of Scalar Functions of a Matrix with respect to a Matrix

      4.5 The Derivative of a Matrix with respect to one of its Elements and Conversely

      4.6 The Derivatives of the Powers of a Matrix

      Problems for Chapter 4

      Chapter 5 – Further Development of Matrix Calculus including an Application of Kronecker Products

      5.1 Introduction

      5.2 Derivatives of Matrices and Kronecker Products

      5.3 The Determination of ( ∂ vec X )/( ∂ vec Y ) for more complicated Equations

      5.4 More on Derivatives of Scalar Functions with respect to a Matrix

      5.5 The Matrix Differential

      Problems for Chapter 5

      Chapter 6 – The Derivative of a Matrix with respect to a Matrix

      6.1 Introduction

      6.2 The Definition and some Results

      6.3 Product Rules for Matrices

      6.4 The Chain Rule for the Derivative of a Matrix with respect to Matrix

      Problems for Chapter 6

      Chapter 7 – Some Applications of Matrix Calculus

      7.1 Introduction

      7.2 The Problems of Least Squares and Constrained Optimization in Scalar Variables

      7.3 Problem 1: Matrix Calculus Approach to the Problems of Least Squares and Constrained Optimization

      7.4 Problem 2: The General Least Squares Problem

      7.5 Problem 3: Maximum Likelihood Estimate of the Multivariate Normal

      7.6 Problem 4: Evaluation of the Jacobians of some Transformations

      7.7 Problem 5: To Find the Derivative of an Exponential Matrix with respect to a Matrix

      Solution to Problems

      Tables of Formulae and Derivatives

      Bibliography

      Index

      Author’s Preface

      My purpose in writing this book is to bring to the attention of the reader, some recent developments in the field of Matrix Calculus. Although some concepts, such as Kronecker matrix products, the vector derivative etc. are mentioned in a few specialised books, no book, to my knowledge, is totally devoted to this subject. The interested researcher must consult numerous published papers to appreciate the scope of the concepts involved.

      Matrix calculus applicable to square matrices was developed by Turnbull [29, 30] as far back as 1927. The theory presented in this book is based on the works of Dwyer and McPhail [15] published in 1948 and others mentioned in the Bibliography. It is more general than Turnbull’s development and is applicable to non-square matrices. But even this more general theory has grave limitations, in particular it requires that in general the matrix elements are non constant and independent. A symmetric matrix, for example, is treated as a special case. Methods of overcoming some of these limitations have been suggested, but I am not aware of any published theory which is both quite general and simple enough to be useful.

      The book is organised in the following way:

      Chapter 1 concentrates on the preliminaries of matrix theory and notation which is found useful throughout the book. In particular, the simple and useful elementary matrix is defined. The vec

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