1st Edition
Matrix Theory From Generalized Inverses to Jordan Form
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
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 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
