| Prerequisite Matrix Theory / 1： |
| Introduction |
| Notation and definitions |
| Inverse |
| Transpose of a matrix |
| Determinants |
| Rank of matrices |
| Quadratic forms |
| Orthogonal matrices |
| Prerequisite Vector Theory / 2： |
| Introduction and definitions |
| Vector space |
| Vector subspaces |
| Linear dependence and independence |
| Basis of a vector space |
| Inner product and orthogonality of vectors |
| Linear Transformations and Characteristic ROOTS / 3： |
| Linear transformations |
| Characteristics roots and vectors |
| Similar matrices |
| Symmetric matrices |
| Geometric Interpretations / 4： |
| Lines in En |
| Planes in En |
| Projections |
| Algebra of Vector Spaces / 5： |
| Intersection and sum of vector spaces |
| Orthogonal complement of a vector subspace |
| Column and null spaces of a matrix |
| Statistical applications |
| Functions of matrices |
| Generalized Inverse; Conditional Inverse / 6： |
| Definition and basic theorems of generalized inverse |
| Systems of linear equations |
| Generalized inverses for special matrices |
| Computing formulas for the g-inverse |
| Conditional inverse |
| Hermite form of matrices |
| Systems of Linear Equations / 7： |
| Existence of solutions to Ax=g |
| the number of solutions of the system Ax=g |
| approximate solutions to inconsistent systems of linear equations |
| Least squares |
| Patterned Matrices And Other Special Matrices / 8： |
| Partitioned matrices |
| The inverse of certain patterned matrices |
| Determinants of certain patterned matrices |
| Characteristic equations and roots of some patterned matrices |
| Triangular matrices |
| Correlation matrix |
| Direct product and sum of matrices |
| Additional theorems |
| Circulants |
| Dominant diagonal matrices |
| Vandermonde and Fourier matrices |
| Permutation matrices |
| Hadamard matrices |
| Band and Toeplitz matrices |
| Trace and Vector of Matrices / 9： |
| Trace |
| Vector of a matrix |
| Commutation matrices |
| Integration and Differentiation / 10： |
| Transformation of random variables |
| Multivariate normal density |
| Moments of density functions and expected values of random matrices |
| Evaluation of a general multiple integral |
| Marginal density function |
| Examples |
| Derivatives |
| Expected values of quadratic forms |
| Expectation of the elements of a Wishart matrix |
| Inverse Positive Matrices and Matrices With Non-Positive Off-Diagonal Elements / 11： |
| Matrices with positive principal minors |
| Matrices with non-positive off-diagonal elements |
| M-matrices (z-matrices with positive principle minors) |
| Z-matrices with non-negative principal minors |
| Non-Negative Matrices; Idempotent and Tripotent Matrices; Projections / 12： |
| Non-negative matrices |
| Idempotent matrices |
| Tripotent matrices |
| Reference and Additional Readings |
| Index |
| Prerequisite Matrix Theory / 1： |
| Introduction |
| Notation and definitions |
| Inverse |
| Transpose of a matrix |
| Determinants |
