Generalizing a decision problem for bipartite perfect matching, Edmonds (J. Res. Natl. Bur. Standards 718(4) (1967) 242) introduced the problem (now known as the Edmonds Problem) of deciding if a given linear subspace ofM(N) contains a non-singular matrix, whereM(N) stands for the linear space of complexN ×N matrices. This problem led to many fundamental developments in matroid theory, etc. Cla...

We prove that the principal pivot transform (also known as partial inverse, sweep operator, or exchange operator in various contexts) maps matrices with positive imaginary part to part. show is matrix monotone by establishing Hermitian square representations for and derivative.

We give elementary proofs of the fact that the Loewner matrices [ f(pi)−f(pj) pi−pj ] corresponding to the function f(t) = t on (0,∞) are positive semidefinite, conditionally negative definite, and conditionally positive definite, for r in [0, 1], [1, 2], and [2, 3], respectively. We show that in contrast to the interval (0,∞) the Loewner matrices corresponding to an operator convex function on...

In this note‎, ‎we obtain some singular values inequalities for positive semidefinite matrices by using block matrix technique‎. ‎Our results are similar to some inequalities shown by Bhatia and Kittaneh in [Linear Algebra Appl‎. ‎308 (2000) 203-211] and [Linear Algebra Appl‎. ‎428 (2008) 2177-2191]‎.

The operator trigonometry of symmetric positive definite (SPD) matrices is extended to arbitrary invertible matrices A and to arbitrary invertible bounded operators A on a Hilbert space. Some background and motivation for these results is provided. © 2000 Elsevier Science Inc. All rights reserved.

Distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive Hermitian matrices. The inverse Fourier transform of the distribution is shown to be a Fredholm determinant of a certain operator that is an analogue of a Wiener-Hopf operator. The asymptotic formula shows that up to the terms of order o(1), the distributions are Gaussian.

Classes of non-Hermitian operators that have only real eigenvalues are presented. Such operators appear in quantum mechanics and are expressed in terms of the generators of the Weyl-Heisenberg algebra. For each non-Hermitian operator A, a Hermitian involutive operator Ĵ such that A is Ĵ-Hermitian, that is, ĴA = AĴ , is found. Moreover, we construct a positive definite Hermitian Q such that A is...

This technical note supplies an affirmative answer to a question raised in a recent pre-print in the context of a “matrix recovery” problem. Assume one samples m Hermitian matrices X1, . . . , Xm with replacement from a finite collection. The deviation of the sum X1+ · · ·+Xm from its expected value in terms of the operator norm can be estimated by an “operator Chernoff-bound” due to Ahlswede a...

A selfadjoint involutive matrix J endows Cn with an indefinite inner product [·, ·] given by [x,y] = 〈Jx,y〉 , x,y ∈ Cn. Characterizations of the J -chaotic order Log(A) J Log(B) are presented for J -selfadjoint matrices A,B with positive eigenvalues, in terms of operator functions involving the α -power mean and the J -relative entropy. An indefinite complete form of the Furuta inequality and s...