Matrix Perturbation Theory
Gilbert W. Stewart, Ji-guang Sun
Academic Press, 1990 - Computers - 365 pages
This book is a comprehensive survey of matrix perturbation theory, a topic of interest to numerical analysts, statisticians, physical scientists, and engineers. In particular, the authors cover perturbation theory of linear systems and least square problems, the eignevalue problem, and the generalized eignevalue problem as wellas a complete treatment of vector and matrix norms, including the theory of unitary invariant norms.
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Norms and Metrics
Linear Systems and Least Squares Problems
The Perturbation of Eigenvalues
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acute perturbation applications assume canonical angles Cmxn Cnxn column space components compute condition number consistent matrix norm convex corresponding defined definite pair denote derive diagonalizable doubly stochastic matrix eigen eigenpair eigenspaces eigenvectors equation equivalent establish example field of values following corollary following theorem Frobenius norm Gerschgorin's theorem Hence Hermitian matrices ill-conditioning invariant subspace inverse Kahan least squares problem Lemma linear systems matrix pairs minimizing Moreover multiple nonnegative nonsingular nonzero norm on Cn normal matrices notation Notes and References orthogonal orthonormal basis partitioned permutation matrix perturbation bounds perturbation theory positive definite projection Proof pseudo-inverse rank(A regular pair relative error residual bounds result satisfying Schur Schur decomposition simple eigenvalue singular value decomposition solution spectral norm Stewart subsection symmetric gauge function Theorem 1.3 topology transformation TZ(Xi unique unitarily invariant norm unitary matrix upper triangular vector norm Wedin xHAx xHBx zero