Papers by Yuji Nakatsukasa
Linear Algebra and Its Applications
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Linear Algebra and Its Applications, 2010
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For standard eigenvalue problems, a closed-form expression for the condition numbers of a multipl... more For standard eigenvalue problems, a closed-form expression for the condition numbers of a multiple eigenvalue is known. In particular, they are uniformly 1 in the Hermitian case, and generally take different values in the non-Hermitian case. We consider the generalized eigenvalue problem and identify the condition numbers of a multiple eigenvalue. Our main result is that a multiple eigenvalue generally has multiple condition numbers, even in the Hermitian definite case. The condition numbers are characterized in terms of the singular values of the outer product of the corresponding left and right eigenvectors.
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We present Gerschgorin-type eigenvalue inclusion sets applicable to generalized eigenvalue proble... more We present Gerschgorin-type eigenvalue inclusion sets applicable to generalized eigenvalue problems.Our sets are defined by circles in the complex plane in the standard Euclidean metric, and are easier to compute than known similar results.As one application we use our results to provide a forward error analysis for a computed eigenvalue of a diagonalizable pencil.
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We prove quadratic eigenvalue perturbation bounds for generalized Hermitian eigenvalue problems. ... more We prove quadratic eigenvalue perturbation bounds for generalized Hermitian eigenvalue problems. The bounds are proportional to the square of the norm of the perturbation matrices divided by the gap between the spectrums. Using the results we provide a simple derivation of the first-order perturbation expansion of a multiple eigenvalue, whose trailing term is tighter than known results. We also present quadratic bounds for the non-Hermitian case.
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For standard eigenvalue problems, a closed-form expression for the condition numbers of a multipl... more For standard eigenvalue problems, a closed-form expression for the condition numbers of a multiple eigenvalue is known. In particular, they are uniformly 1 in the Hermitian case, and generally take different values in the non-Hermitian case. We consider the generalized eigenvalue problem and identify the condition numbers of a multiple eigenvalue. Our main result is that a multiple eigenvalue generally has multiple condition numbers, even in the Hermitian definite case. The condition numbers are characterized in terms of the singular values of the outer product of the corresponding left and right eigenvectors.
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Papers by Yuji Nakatsukasa