Canonical forms for matrix triples (A,G, Ĝ), where A is arbitrary rectangular andG, Ĝare either real symmetric or skew symmetric, or complex Hermitian or skew Hermitian, arederived. These forms generalize classical singular value decompositions. In [1] a similarcanonical form has been obtained for the complex case. In this paper, we provide analternative proof for the comple...

First, we derive explicit computable expressions of structured backward errors of approximate eigenelements of structured matrix polynomials including symmetric, skew-symmetric, Hermitian, skew-Hermitian, even and odd polynomials. We also determine minimal structured perturbations for which approximate eigenelements are exact eigenelements of the perturbed polynomials. Next, we analyze the effe...

This paper is concerned with a generalization of the Hermitian and skew-Hermitian (HSS) splitting iteration for solving positive definite, non-Hermitian linear systems. It is shown that the new scheme can outperform the standard HSS method in some situations and can be used as an effective preconditioner for certain linear systems in saddle point form. Numerical experiments using discretization...

This paper is concerned with a generalization of the Hermitian and skew-Hermitian splitting iteration for solving positive definite, non-Hermitian linear systems. It is shown that the new scheme has some advantages over the standard HSS method, and can be used as an effective preconditioner for certain linear systems in saddle point form. Numerical experiments using discretizations of incompres...

This paper deals with the Hermitian H(A) and skew-Hermitian part S(A) of a complex matrix A. We characterize all complex matrices A such that H(A), respectively S(A), is a potent matrix. Two approaches are used: characterizations of idempotent and tripotent Hermitian matrices of the form [ X Y ∗ Y 0 ] , and a singular value decomposition of A. In addition, a relation between the potency of H(A)...

In quantum mechanics it is well known that observables are represented by hermitian matrices (or operators). Uncertainty relations are represented as some kinds of trace inequalities satisfied by two observables and one density matrix (or operator). Now we try to release the hermitian restriction on observables. This is only a mathematical interest. In this case we give several non-hermitian ex...

This paper deals with the Hermitian H(A) and skew-Hermitian part S(A) of a complex matrix A. We characterize all complex matrices A such that H(A), respectively S(A), is a potent matrix. Two approaches are used: characterizations of idempotent and tripotent Hermitian matrices of the form [ X Y ∗ Y 0 ] , and a singular value decomposition of A. In addition, a relation between the potency of H(A)...