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Matrices with Applications in Statistics (Graybill)

 
 Author(s)  Franklin A. Graybill
 Title  Matrices with Applications in Statistics
 Edition  Second Edition
 Year  1983
 Publisher  Duxbury (Wadsworth Group)
 ISBN  0-534-40131-7
 Website  www.wadsworth.com
 book link
 




Table of Contents


1. PREREQUISITE MATRIX THEORY.
  • 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.

INDEX.




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