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Part of the Duxbury Classic series, Franklin A. Graybill’s MATRICES WITH APPLICATIONS TO STATISTICS focuses primarily on matrices as they relate to areas of multivariate analysis and the linear model. This seminal work is a time tested, authoritative resource for both students and researchers.
Introduction. Notation and definitions. Inverse. Transpose of a matrix. Determinants. Rank of matrices. Quadratic forms. Orthogonal matrices.
2. PREREQUISITE VECTOR THEORY.
Introduction and definitions. Vector space. Vector subspaces. Linear dependence and independence. Basis of a vector space. Inner product and orthogonality of vectors.
3. LINEAR TRANSFORMATIONS AND CHARACTERISTIC ROOTS.
Linear transformations. Characteristics roots and vectors. Similar matrices. Symmetric matrices.
4. GEOMETRIC INTERPRETATIONS.
Introduction. Lines in En. Planes in En. Projections.
5. ALGEBRA OF VECTOR SPACES.
Introduction. Intersection and sum of vector spaces. Orthogonal complement of a vector subspace. Column and null spaces of a matrix. Statistical applications. Functions of matrices.
6. GENERALIZED INVERSE; CONDITIONAL INVERSE.
Introduction. 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.
7. SYSTEMS OF LINEAR EQUATIONS.
Introduction. Existence of solutions to Ax=g. the number of solutions of the system Ax=g. approximate solutions to inconsistent systems of linear equations. Statistical applications. Least squares. Statistical applications.
8. PATTERNED MATRICES AND OTHER SPECIAL MATRICES.
Introduction. 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.
9. TRACE AND VECTOR OF MATRICES.
Trace. Vector of a matrix. Commutation matrices.
10. INTEGRATION AND DIFFERENTIATION.
Introduction. 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.
11. INVERSE POSITIVE MATRICES AND MATRICES WITH NON-POSITIVE OFF-DIAGONAL ELEMENTS.
Introduction and definitions. 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.
12. NON-NEGATIVE MATRICES; IDEMPOTENT AND TRIPOTENT MATRICES; PROJECTIONS.
Introduction. Non-negative matrices. Idempotent matrices. Tripotent matrices. Projections. Additional theorems.
REFERENCE AND ADDITIONAL READINGS.