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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 | |