1st Edition

# Matrix Theory From Generalized Inverses to Jordan Form

By Robert Piziak, P.L. Odell Copyright 2007
568 Pages 27 B/W Illustrations
by Chapman & Hall

568 Pages 27 B/W Illustrations
by Chapman & Hall

568 Pages
by Chapman & Hall

Also available as eBook on:

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.

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 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
Scalar Multiplication
Matrix Multiplication
Transpose
Submatrices

APPENDIX C: DETERMINANTS
Motivation
Defining Determinants
The Trace of a Square Matrix

APPENDIX D: A REVIEW OF BASICS
Spanning
Linear Independence
Basis and Dimension
Change of Basis

INDEX

### Biography

Piziak, Robert; Odell, P.L.

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 2011

Always 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