Advanced Matrix Theory

Overview

Contents:
Introduction : First Basic Problem Systems of Linear equations - Matrix Notation The various questions that arise with a system of linear eqautions - Second Basic Problem Diagonalization of a square matrix The various questions that arise with diagonalization.

Vector Spaces : Vector spaces - Subspaces - Linear combinations and subspaces spanned by a set of vectors - Linear dependence and Linear independence - Spanning Set and Basis - Finite dimensional spaces - Dimension

Solutions of Linear Systems : Simple systems - Homogeneous and Nonhomogeneous systems - Gaussian elimination - Null Space and Range - Rank and nullity - Consistency conditions in terms of rank - General Solution of a linear system - Elementary Row and Column operations - Row Reduced Form - Triangular Matrix Factorization

Important Subspaces associsted with a matrix : Range and Null space - Rank and Nullity - Rank Nullity theorem - Four Fundamental subspaces - Orientation of the four subspaces

Orthogonality : Inner product - Inner product Spaces - Cauchy Schwarz inequality - Norm - Orthogonality - Gram Schmidt orthonormalization - Orthonormal basis - Expansion in terms of orthonormal basis Fourier series
Orthogonal complement - Decomposition of a vector with respect to a subspace and its orthogonal - complement Pythagorus Theorem

Eigenvalues and Eigenvectors : What are the ingredients required for diagonalization? - Eigenvalue Eigenvector pairs - Where do we look for eigenvalues? characteristic equation - Algebraic multiplicity - Eigenvectors, Eigenspaces and geometric multiplicity

Diagonalizable Matrices : Diagonalization criterion - The diagonalizing matrix - Cayley-Hamilton theorem, Annihilating polynomials, Minimal Polynomial - Diagonalizability and Minimal polynomial - Projections - Decomposition of the matrix in terms of projections

Hermitian Matrices : Real symmetric and Hermitian Matrices - Properties of eigenvalues and eigenvectors - Unitary/Orthoginal Diagonalizbility of Complex Hermitian/Real Symmetric - matrices - Spectral Theorem - Positive and Negative Definite and Semi definite matrices

General Matrices : The matrices AAT and ATA - Rank, Nullity, Range and Null Space of AAT and ATA - Strategy for choosing the basis for the four fundamental subspaces - Singular Values - Singular Value Decomposition - Pseudoinverse and Optimal solution of a linear system of equations - The Geometry of Pseudoinverse

Jordan Cnonical form* : Primary Decomposition Theorem - Nilpotent matrices - Canonical form for a nilpotent matrix - Jordan Canonical Form - Functions of a matrix.

Selected Topics in Applications* : Optimization and Linear Programming - Network models - Game Theory - Control Theory - Image Compression