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Matrix Theory

From Generalized Inverses to Jordan Form

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## Book Description

In 1990, the National Science Foundation recommended that every college mathematics curriculum should include a second course in linear algebra. In answer to this recommendation, **Matrix Theory: From Generalized Inverses to Jordan Form** provides the material for a second semester of linear algebra that probes introductory linear algebra concepts while also exploring topics not typically covered in a sophomore-level class.

Tailoring the material to advanced undergraduate and beginning graduate students, the authors offer instructors flexibility in choosing topics from the book. The text first focuses on the central problem of linear algebra: solving systems of linear equations. It then discusses LU factorization, derives Sylvester's rank formula, introduces full-rank factorization, and describes generalized inverses. After discussions on norms, QR factorization, and orthogonality, the authors prove the important spectral theorem. They also highlight the primary decomposition theorem, Schur's triangularization theorem, singular value decomposition, and the Jordan canonical form theorem. The book concludes with a chapter on multilinear algebra.

With this classroom-tested text students can delve into elementary linear algebra ideas at a deeper level and prepare for further study in matrix theory and abstract algebra.

## Table of Contents

THE IDEA OF INVERSE

Solving Systems of Linear Equations

The Special Case of "Square" Systems

GENERATING INVERTIBLE MATRICES

A Brief Review of Gauss Elimination with Back Substitution

Elementary Matrices

The LU and LDU Factorization

The Adjugate of a Matrix

The Frame Algorithm and the Cayley-Hamilton Theorem

SUBSPACES ASSOCIATED TO MATRICES

Fundamental Subspaces

A Deeper Look at Rank

Direct Sums and Idempotents

The Index of a Square Matrix

Left and Right Inverses

THE MOORE-PENROSE INVERSE

Row Reduced Echelon Form and Matrix Equivalence

The Hermite Echelon Form

Full Rank Factorization

The Moore-Penrose Inverse

Solving Systems of Linear Equations

Schur Complements Again

GENERALIZED INVERSES

The {1}-Inverse

{1,2}-Inverses

Constructing Other Generalized Inverses

{2}-Inverses

The Drazin Inverse

The Group Inverse

NORMS

The Normed Linear Space Cn

Matrix Norms

INNER PRODUCTS

The Inner Product Space Cn

Orthogonal Sets of Vectors in Cn

QR Factorization

A Fundamental Theorem of Linear Algebra

Minimum Norm Solutions

Least Squares

PROJECTIONS

Orthogonal Projections

The Geometry of Subspaces and the Algebra of Projections

The Fundamental Projections of a Matrix

Full Rank Factorizations of Projections

Affine Projections

Quotient Spaces

SPECTRAL THEORY

Eigenstuff

The Spectral Theorem

The Square Root and Polar Decomposition Theorems

MATRIX DIAGONALIZATION

Diagonalization with Respect to Equivalence

Diagonalization with Respect to Similarity

Diagonalization with Respect to a Unitary

The Singular Value Decomposition

JORDAN CANONICAL FORM

Jordan Form and Generalized Eigenvectors

The Smith Normal Form

MULTILINEAR MATTERS

Bilinear Forms

Matrices Associated to Bilinear Forms

Orthogonality

Symmetric Bilinear Forms

Congruence and Symmetric Matrices

Skew-Symmetric Bilinear Forms

Tensor Products of Matrices

APPENDIX A: COMPLEX NUMBERS

What is a Scalar?

The System of Complex Numbers

The Rules of Arithmetic in C

Complex Conjugation, Modulus, and Distance

The Polar Form of Complex Numbers

Polynomials over C

Postscript

APPENDIX B: BASIC MATRIX OPERATIONS

Introduction

Matrix Addition

Scalar Multiplication

Matrix Multiplication

Transpose

Submatrices

APPENDIX C: DETERMINANTS

Motivation

Defining Determinants

Some Theorems about Determinants

The Trace of a Square Matrix

APPENDIX D: A REVIEW OF BASICS

Spanning

Linear Independence

Basis and Dimension

Change of Basis

INDEX

## Author(s)

### Biography

Piziak, Robert; Odell, P.L.

## Reviews

Each chapter ends with a list of references for further reading. Undoubtedly, these will be useful for anyone who wishes to pursue the topics deeper. … the book has many MATLAB examples and problems presented at appropriate places. … the book will become a widely used classroom text for a second course on linear algebra. It can be used profitably by graduate and advanced level undergraduate students. It can also serve as an intermediate course for more advanced texts in matrix theory. This is a lucidly written book by two authors who have made many contributions to linear and multilinear algebra.

—K.C. Sivakumar,IMAGE, No. 47, Fall 2011Always mathematically constructive, this book helps readers delve into elementary linear algebra ideas at a deeper level and prepare for further study in matrix theory and abstract algebra.

—L’enseignement Mathématique, January-June 2007, Vol. 53, No. 1-2