In an earlier paper [GKO95] we exploited the displacement structure of Cauchy-like matrices to derive for them a fast O(n) implementation of Gaussian elimination with partial pivoting. One application is to the rapid and numerically accurate solution of linear systems with Toeplitzlike coe cient matrices, based on the fact that the latter can be transformed into Cauchy-like matrices by using th...

Two Hermitian matrices A, B ∈ Mn(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C ∈ Mn(C) such that B = CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible iner...

For solving large sparse non-Hermitian positive definite linear equations, Bai et al. proposed the Hermitian and skew-Hermitian splitting methods (HSS). They recently generalized this technique to the normal and skew-Hermitian splitting methods (NSS). In this paper, we present an accelerated normal and skew-Hermitian splitting methods (ANSS) which involve two parameters for the NSS iteration. W...

In this paper, we first construct a preconditioned two-parameter generalized Hermitian and skew-Hermitian splitting (PTGHSS) iteration method based on the two-parameter generalized Hermitian and skew-Hermitian splitting (TGHSS) iteration method for non-Hermitian positive definite linear systems. Then a class of PTGHSSbased iteration methods are proposed for solving weakly nonlinear systems base...

The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we give various closed-form formulas for the maximal and minimal values for the rank and inertia of the Hermitian expression A + X, where A is a given Hermitian matrix and X is a variable Hermi...

Non-Hermitian band theory distinguishes between line gaps and point gaps. While can give rise to intrinsic non-Hermitian topology without Hermitian counterparts, line-gapped systems always be adiabatically deformed a limit. Here we show that line-gap point-gap intricately connected: topological in $d$ dimensions induce nontrivial on their $(d-1)$-dimensional boundaries when suitable internal sp...

Non-Hermitian topological edge states have many intriguing properties, but so far mainly been discussed in terms of bulk-boundary correspondence. Here we propose to use a bulk property diffusion coefficients for probing the and exploring their dynamics. The coefficient is found show unique features with phase transitions driven by paritytime( PT)-symmetric non-Hermitian discrete-time quantum wa...

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields over Q( √ −m) for all m. For each imaginary quadratic field Q( √ −m), we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note t...

The paper deal with non-commutative geometry. The notion of quantumspheres was introduced by podles. Here we define the quantum hermitianmetric on the quantum spaces and find it for the quantum spheres.