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 ProblemsDefinitions 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 ProblemsUnique solution of linear equations Two equations and unknowns Gaussian elimination Triangular decomposition Ill-conditioning ProblemsDeterminant and inverse 3x3 case General properties Some applications Evaluation of determinants Definition and properties Partitioned form Calculation of inverse Cramer’s rule ProblemsUnique 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 ProblemsDefinitions 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 ProblemsQuadratic and hermitian forms Definitions Lagrange’s reduction of quadratic forms Sylvester’s law of inertia Definitions Tests Optimization of functions Rayleigh quotient Liapunov stability ProblemsCanonical forms Jordan form Normal forms Schur form Hessenberg form Singular value and polar decompositions ProblemsDefinition and properties Sylvester’s formula Linear differential and difference equations Matrix sign function ProblemsDefinition Properties Computation Other inverses (i, j, k) inverses Drazin inverse Solution of linear equations Linear feedback control Singular systems Estimation of parameters ProblemsCompanion 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 ProblemsPolynomial and rational matrices Basic properties of polynomial matrices Elementary operations and Smith normal form Relative primeness Skew primeness Smith-McMillan form Transfer function matrices ProblemsPatterned matrices Banded matrices Circulant matrices Toeplitz and Hankel matrices Brownian matrices Centrosymmetric matrices Comrade matrix Loewner matrix Permutation matrices Sequence Hankel matrices ProblemsAX = 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 matricesExercisesProblems References for chapters Some additional references on applications Answers to exercises Answers to problems Index
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