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# Barnett S. Matrices methods and applications

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Oxford: Oxford University Press, 1996. - 466p.
This volume provides a down-to-earth, easily understandable guide to techniques of matrix theory, which are widely used throughout engineering and the physical, life, and social sciences. Fully up-to-date, the book covers a wide range of topics, from basic matrix algebra to such advanced concepts as generalized inverses and Hadamard matrices, and applications to error-correcting codes, control theory, and linear programming. Results are illustrated with many examples drawn from diverse areas of application. Numerous exercises are included to clarify the material presented in the text, which is suitable for undergraduates and graduates alike. Researchers will also benefit from the accessible accounts of advanced matrix techniques.
Contents:
Preface
Notation
How matrices arise
Problems
Definitions
Multiplication by a scalar
Multiplication of two matrices
Definition and properties
Symmetric and hermitian matrices
Partitioning and submatrices
Kronecker and Hadamard products
Derivative of a matrix
Problems
Unique solution of linear equations
Two equations and unknowns
Gaussian elimination
Triangular decomposition
Ill-conditioning
Problems
Determinant and inverse
3x3 case
General properties
Some applications
Evaluation of determinants
Definition and properties
Partitioned form
Calculation of inverse
Cramer’s rule
Problems
Unique solution
Definition of rank
Elementary operations
Calculation of rank
Normal form
Homogeneous equations
Inhomogeneous equations
Consistency theorem
Method of least squares
Use of Kronecker product
Linear dependence of vectors
Error-correcting codes
Problems
Definitions
Some applications
The characteristic equation
Hermitian and symmetric matrices
Matrix polynomials and the Cayley-Hamilton theorem
Companion matrix
Kronecker product expressions
Definition
Diagonalization
Hermitian and symmetric matrices
Transformation to companion form
Solution of linear differential and difference equations
Power method
Other methods
Gauss-Seidel and Jacobi methods
Newton-Raphson type method
Problems
Quadratic and hermitian forms
Definitions
Lagrange’s reduction of quadratic forms
Sylvester’s law of inertia
Definitions
Tests
Optimization of functions
Rayleigh quotient
Liapunov stability
Problems
Canonical forms
Jordan form
Normal forms
Schur form
Hessenberg form
Singular value and polar decompositions
Problems
Definition and properties
Sylvester’s formula
Linear differential and difference equations
Matrix sign function
Problems
Definition
Properties
Computation
Other inverses
(i, j, k) inverses
Drazin inverse
Solution of linear equations
Linear feedback control
Singular systems
Estimation of parameters
Problems
Companion matrices
Resultant matrices
Sylvester matrix
Bezoutian matrix
Computation via row operations
Euclid’s algorithm
Diophantine equations
Relative to the imaginary axis
Relative to the unit circle
Bilinear transformation
Liapunov equations
Riccati equation
Solution via eigenvectors
Solution via matrix sign function
Problems
Polynomial and rational matrices
Basic properties of polynomial matrices
Elementary operations and Smith normal form
Relative primeness
Skew primeness
Smith-McMillan form
Transfer function matrices
Problems
Patterned matrices
Banded matrices
Circulant matrices
Toeplitz and Hankel matrices
Brownian matrices
Centrosymmetric matrices
Comrade matrix
Loewner matrix
Permutation matrices
Sequence Hankel matrices
Problems
AX = XB
Commuting matrices
f(X) = 0
Other equations
Basic properties
M-matrices
Stochastic matrices
Other forms
Vector norms
Matrix norms
Conditioning
Hadamard matrices
Inequalities
Interval matrices
Unimodular integer matrices
Exercises
Problems
References for chapters
Some additional references on applications
Answers to exercises
Answers to problems
Index
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