O'Dorney, Evan
Published in
Linear Algebra and Its Applications

The Cayley transform, $(A)=(I−A)(I+A)−1, maps skew-symmetric matrices to orthogonal matrices and vice versa. Given an orthogonal matrix Q, we can choose a diagonal matrix D with each diagonal entry ±1 (a signature matrix) and, if I+QD is nonsingular, calculate the skew-symmetric matrix $(QD). An open problem is to show that, by a suitable choice of...

de Seguins Pazzis, Clément
Published in
Linear Algebra and Its Applications

Let K be a (commutative) field with characteristic not 2, and V be a linear subspace of matrices of Mn(K) that have at most two eigenvalues in K (respectively, at most one non-zero eigenvalue in K). We prove that dimV⩽(n2)+2 provided that n⩾3 (respectively, dimV⩽(n2)+1).We also classify, up to similarity, the linear subspaces of Mn(K) in which ever...

Ahmadi, S. Ruhallah Izadi, Mohammad A. Szechtman, Fernando
Published in
Linear Algebra and Its Applications

All bilinear forms defined on a finite dimensional vector space over an algebraically closed field of characteristic 2 whose associated Lie algebra is reductive are determined.

Kalita, Debajit Pati, Sukanta
Published in
Linear Algebra and Its Applications

A weighted directed graph is a directed graph G whose underlying undirected graph is simple and whose edges have nonzero (directional) complex weights, that is, the presence of an edge (u,v) of weight w is as good as the presence of the edge (v,u) with weight w¯, the complex conjugate of w. Let G be a weighted directed graph on vertices 1,2,…,n. De...

Hou, Jinchuan Li, Chi-Kwong Poon, Yiu-Tung Qi, Xiaofei Sze, Nung-Sing
Published in
Linear Algebra and Its Applications

We study k-positive maps on operators. We obtain a new criterion on k-positivity in terms of the k-numerical range, and use it to improve and refine some earlier results on k-positive maps related to the study of quantum information science. We also consider a special class of positive maps extending the construction of Choi on positive maps that a...

Zheng, Baodong Xu, Jinli Fošner, Ajda
Published in
Linear Algebra and Its Applications

Let F be a field of characteristic not 2 and Mn the algebra of all n×n matrices over F. The aim of this paper is to characterize linear maps ϕ:Mm1⋯ml→Mm1⋯ml such that ϕ(A1⊗⋯⊗Al) is an idempotent whenever A1⊗⋯⊗Al is an idempotent.

Brualdi, Richard A. Fritscher, Eliseu
Published in
Linear Algebra and Its Applications

After considerable discussion intended to elucidate the connections between permutation matrices and their Hankel and Toeplitz X-rays, tournaments, transversals of partial Latin squares, and Skolem sequences, we prove several theorems concerning the existence of permutation matrices whose Hankel and Toeplitz X-rays have special properties such as b...

van der Woude, Jacob
Published in
Linear Algebra and Its Applications

In this paper we present a straightforward proof of the well-known fact that a positive semi-definite polynomial matrix can be written as the product of a polynomial matrix and its associated (Hermitian) transposed. For polynomial matrices with complex coefficients both factors also have complex coefficients and can be taken of the same size as the...

Ma, Chao
Published in
Linear Algebra and Its Applications

We first characterize idempotent zero patterns. Then we determine the possible numbers of nonzero entries in an idempotent zero pattern with a given minimum rank and characterize those patterns that attain the extremal numbers. The results can be stated in terms of 0–1 matrices.

Ghebleh, M.
Published in
Linear Algebra and Its Applications

Let A(R,S) denote the class of all matrices of zeros and ones with row sum vector R and column sum vector S. We introduce the notion of an inversion in a (0,1)-matrix. This definition extends the standard notion of an inversion of a permutation, in the sense that both notions agree on the class of permutation matrices. We prove that the number of i...