**Matrices, Vectors, and Their Operations**

Basic definitions and notations

Matrix addition and scalar-matrix multiplication

Matrix multiplication

Partitioned matrices

The "trace" of a square matrix

Some special matrices

**Systems of Linear Equations**

Introduction

Gaussian elimination

Gauss-Jordan elimination

Elementary matrices

Homogeneous linear systems

The inverse of a matrix

**More on Linear Equations**

The *LU* decomposition

Crout’s Algorithm

*LU* decomposition with row interchanges

The *LDU* and Cholesky factorizations

Inverse of partitioned matrices

The *LDU* decomposition for partitioned matrices

The Sherman-Woodbury-Morrison formula

**Euclidean Spaces**

Introduction

Vector addition and scalar multiplication

Linear spaces and subspaces

Intersection and sum of subspaces

Linear combinations and spans

Four fundamental subspaces

Linear independence

Basis and dimension

**The Rank of a Matrix**

Rank and nullity of a matrix

Bases for the four fundamental subspaces

Rank and inverse

Rank factorization

The rank-normal form

Rank of a partitioned matrix

Bases for the fundamental subspaces using the rank normal form

**Complementary Subspaces**

Sum of subspaces

The dimension of the sum of subspaces

Direct sums and complements

Projectors

**Orthogonality, Orthogonal Subspaces, and Projections**

Inner product, norms, and orthogonality

Row rank = column rank: A proof using orthogonality

Orthogonal projections

Gram-Schmidt orthogonalization

Orthocomplementary subspaces

The fundamental theorem of linear algebra

**More on Orthogonality**

Orthogonal matrices

The *QR* decomposition

Orthogonal projection and projector

Orthogonal projector: Alternative derivations

Sum of orthogonal projectors

Orthogonal triangularization

**Revisiting Linear Equations**

Introduction

Null spaces and the general solution of linear systems

Rank and linear systems

Generalized inverse of a matrix

Generalized inverses and linear systems

The Moore-Penrose inverse

**Determinants**

Definitions

Some basic properties of determinants

Determinant of products

Computing determinants

The determinant of the transpose of a matrix — revisited

Determinants of partitioned matrices

Cofactors and expansion theorems

The minor and the rank of a matrix

The Cauchy-Binet formula

The Laplace expansion

**Eigenvalues and Eigenvectors**

Characteristic polynomial and its roots

Spectral decomposition of real symmetric matrices

Spectral decomposition of Hermitian and normal matrices

Further results on eigenvalues

Singular value decomposition

**Singular Value and Jordan Decompositions **

Singular value decomposition (SVD)

The SVD and the four fundamental subspaces

SVD and linear systems

SVD, data compression and principal components

Computing the SVD

The Jordan canonical form

Implications of the Jordan canonical form

**Quadratic Forms**

Introduction

Quadratic forms

Matrices in quadratic forms

Positive and nonnegative definite matrices

Congruence and Sylvester’s law of inertia

Nonnegative definite matrices and minors

Extrema of quadratic forms

Simultaneous diagonalization

**The Kronecker Product and Related Operations **

Bilinear interpolation and the Kronecker product

Basic properties of Kronecker products

Inverses, rank and nonsingularity of Kronecker products

Matrix factorizations for Kronecker products

Eigenvalues and determinant

The vec and commutator operators

Linear systems involving Kronecker products

Sylvester’s equation and the Kronecker sum

The Hadamard product

**Linear Iterative Systems, Norms, and Convergence **

Linear iterative systems and convergence of matrix powers

Vector norms

Spectral radius and matrix convergence

Matrix norms and the Gerschgorin circles

SVD – revisited

Web page ranking and Markov chains

Iterative algorithms for solving linear equations

**Abstract Linear Algebra **

General vector spaces

General inner products

Linear transformations, adjoint and rank

The four fundamental subspaces - revisited

Inverses of linear transformations

Linear transformations and matrices

Change of bases, equivalence and similar matrices

Hilbert spaces

**References**