On spectral condition of J-Herminian operators (original) (raw)
2000, Glasnik Matematicki
The spectral condition of a matrix H is the inm um of the condition numbers (Z) = kZkkZ 1 k, taken over all Z such that Z 1 HZ is diagonal. This number controls the sensitivity of the spectrum of H under perturbations. A matrix is called J-Hermitian if H = JHJ for some J = J = J 1 . When diagonalizing J-Hermitian matrices it is natural to look at J-unitary Z, that is, those that satisfy Z JZ = J. Our rst result is: if there is such J-unitary Z, then the inm um above is taken on J-unitary Z, that is, the J unitary diagonalization is the most stable of all. For the special case when J-Hermitian matrix has denite spectrum, we give various upper bounds for the spectral condition, and show that all J-unitaries Z which diagonalize such a matrix have the same condition number. Our estimates are given in the spectral norm and the Hilbert{Schmidt norm. Our results are, in fact, formulated and proved in a general Hilbert space (under an appropriate generalization of the notion of 'diagona...
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On spectral condition of J-Hermitian operators
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Let J be a fixed real or complex n×n matrix such that J=J * =J -1 · A matrix H is called J-Hermitian if H * =JHJ; or, equivalently, JH is Hermitian. A matrix U is called J-unitary if U * JU=J. If H is diagonalizable, then we define the spectral condition number for H to be the infimum of the condition numbers κ(Z) taken over all Z such that Z -1 HZ is diagonal. The authors show (Theorem 3.3) that if a J-Hermitian matrix H is diagonalizable by some J-unitary matrix (in general this need not happen), then the spectral condition number is attained by some J-unitary matrix Z· They also prove (Theorem 4.1) that if JH is positive definite and there exists a J-unitary matrix U such that U -1 HU is Hermitian (as well as J-Hermitian), then κ(U)≤κ(D * JHD) for all nonsingular matrices D which commute with J· Applications are made to Klein-Gordon operators. The results as stated here can be generalized to infinite dimensional space; in the paper they are stated and proved for general Hilbert s...
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