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