Matrix Perturbation Theory
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.
Generalized Eigenvalue Problems
Norms and Metrics
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A₁ acute perturbation approximation canonical angles Cmxn Cnxn column space components compute condition number consistent matrix norm convex corresponding defined definite pair denote disks doubly stochastic matrix eigen eigenpair eigenspaces eigenvalue problem eigenvalues of Ã eigenvectors equation equivalent establish example field of values following theorem Frobenius norm G. W. Stewart Gerschgorin Hence Hermitian matrices ill-conditioning invariant subspace inverse L₁ least squares problem Lemma linear systems matrix pairs Moreover nonnegative nonsingular nonzero normal notation Notes and References orthogonal orthonormal basis P₁ permutation matrix perturbation bounds perturbation theory positive definite projection Proof pseudo-inverse rank(A re(A relative error residual bounds result satisfies Schur Schur decomposition Show singular value decomposition solution spectral norm subsection symmetric gauge function transformation unique unitarily invariant norms unitary matrix upper triangular V₁ vector norm Wedin X₁ XHAX Y₁ zero