Matrix Theory

Piziak, Robert; Odell, P.L.

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