Matrix Theory

ISBN-10: 0486411796
ISBN-13: 9780486411798
Edition: N/A
Authors: Joel N. Franklin
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Description: Solid, mathematically rigorous introduction covers diagonalizations and triangularizations of Hermitian and non-Hermitian matices, the matrix theorem of Jordan, variational principles and perturbation theory of matrices, matrix numerical analysis,  More...

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Book details

List price: $16.95
Publisher: Dover Publications, Incorporated
Publication date: 2/8/2000
Binding: Paperback
Pages: 320
Size: 5.50" wide x 9.00" long x 0.75" tall
Weight: 0.682
Language: English

Solid, mathematically rigorous introduction covers diagonalizations and triangularizations of Hermitian and non-Hermitian matices, the matrix theorem of Jordan, variational principles and perturbation theory of matrices, matrix numerical analysis, in-depth analysis of linear computations, more. Only elementary algebra and calculus. Problem-solving exercises. 1968 edition.

Determinants
Introduction
The Definition of a Determinant
Properties of Determinants
Row and Column Expansions
Vectors and Matrices
The Inverse Matrix
The Determinant of a Matrix Product
The Derivative of a Determinant
The Theory of Linear Equations
Introduction
Linear Vector Spaces
Basis and Dimension
Solvability of Homogeneous Equations
Evaluation of Rank by Determinants
The General m x n Inhomogeneous System
Least-Squares Solution of Unsolvable Systems
Matrix Analysis of Differential Equations
Introduction
Systems of Linear Differential Equations
Reduction to the Homogeneous System
Solution by the Exponential Matrix
Solution by Eigenvalues and Eigenvectors
Eigenvalues, Eigenvectors, and Canonical Forms
Matrices with Distinct Eigenvalues
The Canonical Diagonal Form
The Trace and Other Invariants
Unitary Matrices
The Gram-Schmidt Orthogonalization Process
Principal Axes of Ellipsoids
Hermitian Matrices
Mass-spring Systems
Positive Definiteness
Simultaneous Diagonalization
Unitary Triangularization
Normal Matrices
The Jordan Canonical Form
Introduction
Principal Vectors
Proof of Jordan's Theorem
Variational Principles and Perturbation Theory
Introduction
The Rayleigh Principle
The Courant Minimax Theorem
The Inclusion Principle
A Determinant-criterion for Positive Definiteness
Determinants as Volumes
Hadamard's Inequality
Weyl's Inequalities
Gershgorin's Theorem
Vector Norms and the Related Matrix Norms
The Condition-Number of a Matrix
Positive and Irreducible Matrices
Perturbations of the Spectrum
Continuous Dependence of Eigenvalues on Matrices
Numerical Methods
Introduction
The Method of Elimination
Factorization by Triangular Matrices
Direct Solution of Large systems of Linear Equations
Reduction of Rounding Error
The Gauss-Seidel and Other Iterative Methods
Computation of Eigenvectors from Known Eigenvalues
Numerical Instability of the Jordan Canonical Form
The Method of Iteration for Dominant Eigenvalues
Reduction to Obtain the Smaller Eigenvalues
Eigenvalues and Eigenvectors of Tridiagonal and Hessenberg Matrices
The Method of Householder and Bauer
Numerical Identification of Stable Matrices
Accurate Unitary Reduction to Triangular Form
The QR Method for Computing Eigenvalues
Index

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