On spectral condition of J-Herminian operators

The spectral condition of a matrix H is the infimum of the condition numbers κ(Z) = ||Z|| ||Z -1||, taken over all Z such that Z -1HZ 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 first result is: if there is such J-unitary Z, then the infimum 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 definite 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 \u27diagonalising\u27) and they are applicable even to unbounded operators. We apply our theory to the Klein-Gordon operator thus improving a previously known bound

On spectral condition of J-Herminian operators

The spectral condition of a matrix H is the infimum of the condition numbers κ(Z) = ||Z|| ||Z -1||, taken over all Z such that Z -1HZ 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 first result is: if there is such J-unitary Z, then the infimum 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 definite 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 \u27diagonalising\u27) and they are applicable even to unbounded operators. We apply our theory to the Klein-Gordon operator thus improving a previously known bound

Computing the singular value decomposition with high relative accuracy

AbstractWe analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, which in general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as finite element problems and quantum mechanics, it is the smallest singular values that have physical meaning, and should be determined accurately by the data. Many recent papers have identified special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite different, motivating us to seek a common perturbation theory and common algorithm. We provide these in this paper, and show that high relative accuracy is possible in many new cases as well. The briefest way to describe our results is that we can compute the SVD of G to high relative accuracy provided we can accurately factor G=XDYT where D is diagonal and X and Y are any well-conditioned matrices; furthermore, the LDU factorization frequently does the job. We provide many examples of matrix classes permitting such an LDU decomposition

Representation Theorems for Indefinite Quadratic Forms Revisited

The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring the second representation theorem to hold is proved. A new simple and explicit example of a self-adjoint operator for which the second representation theorem does not hold is also provided.Comment: This work is supported in part by the Deutsche Forschungsgemeinschaf