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On low-dimensional partial isometries
No full textTwo statements concerning -by- partial isometries are being considered: (i) these matrices are generic, if unitarily irreducible, and (ii) if nilpotent, their numerical ranges are circular disks. Both statements hold for but fail starting with
Preservers of the p-power and the Wasserstein means on 2x2 matrices
No full textIn one of his recent papers, Molnár showed that if is a von Neumann algebra without -type direct summands, then any function from the positive definite cone of to the positive real numbers preserving the Kubo-Ando power mean, for some is necessarily constant. It was shown in that paper that -type algebras admit nontrivial -power mean preserving functionals, and it was conjectured that -type algebras admit only constant -power mean preserving functionals. We confirm the latter. A similar result occurred in another recent paper of Molnár concerning the Wasserstein mean. We prove the conjecture for -type algebras in regard of the Wasserstein mean, too. We also give two conditions that characterise centrality in -algebras
Products of skew-involutions
No full textIt is shown that every -by- matrix over a field with determinant 1 is a product of (i) four or fewer skew-involutions () provided , and (ii) eight or fewer skew-involutions if and n > 1. Every real symplectic matrix is a product of six real symplectic skew-involutions, and an explicit factorization of a complex symplectic matrix into two symplectic skew-involutions is given
Hypocoercivity and hypocontractivity concepts for linear dynamical systems
No full textFor linear dynamical systems (in continuous-time and discrete-time), we revisit and extend the concepts of hypocoercivity and hypocontractivity and give a detailed analysis of the relations of these concepts to (asymptotic) stability, as well as (semi-)dissipativity and (semi-)contractivity, respectively. On the basis of these results, the short-time behavior of the propagator norm for linear continuous-time and discrete-time systems is characterized by the (shifted) hypocoercivity index and the (scaled) hypocontractivity index, respectively
Extending CSR decomposition to tropical inhomogeneous matrix products
No full textThis article presents an attempt to extend the CSR decomposition, previously introduced for tropical matrix powers, to tropical inhomogeneous matrix products. The CSR terms for inhomogeneous matrix products are introduced, and then, a case is described where an inhomogeneous product admits such CSR decomposition after some length and a bound on this length is given. In the last part of the paper, a number of counterexamples are presented to show that inhomogeneous products do not admit CSR decomposition under more general conditions
The Graham-Hoffman-Hosoya-type theorems for the exponential distance matrix
No full textLet be a strongly connected digraph with vertex set . Denote by the distance between vertices and in . Two variant versions of the distance matrix were proposed by Yan and Yeh (Adv. Appl. Math.), and Bapat et al. (Linear Algebra Appl.) independently, one is the -distance matrix, and the other is the exponential distance matrix. Given a nonzero indeterminate , the -distance matrix of is defined asIn particular, when , it would be reduced to the distance matrix of . The exponential distance matrix of is defined as In , Graham et al. (J. Graph Theory) established a classical formula connecting the determinants and cofactor sums of the distance matrices of strongly connected digraphs in terms of their blocks, which plays a powerful role in the subsequent researches on the determinants of distance matrices. Sivasubramanian (Electron. J. Combin.) and Li et al. (Discuss. Math. Graph Theory) independently extended it from the distance matrix to the -distance matrix. In this note, three formulae of such types for the exponential distance matrices of strongly connected digraphs will be presented
Orthogonal realizations of random sign patterns and other applications of the SIPP
No full textA sign pattern is an array with entries in . A real matrix is row orthogonal if . The Strong Inner Product Property (SIPP), introduced in [B.A. Curtis and B.L. Shader, Sign patterns of orthogonal matrices and the strong inner product property, Linear Algebra Appl. 592: 228-259, 2020], is an important tool when determining whether a sign pattern allows row orthogonality because it guarantees there is a nearby matrix with the same property, allowing zero entries to be perturbed to nonzero entries, while preserving the sign of every nonzero entry. This paper uses the SIPP to initiate the study of conditions under which random sign patterns allow row orthogonality with high probability. Building on prior work, nowhere zero sign patterns that minimally allow orthogonality are determined. Conditions on zero entries in a sign pattern are established that guarantee any row orthogonal matrix with such a sign pattern has the SIPP
Semipositivity with respect to the Lorentz cone
No full textLorentz cone in the Euclidean space is defined as . The paper aims to study semipositivity of matrices with respect to . A real matrix is -semipositive if there exists such that (the topological interior of ). -positive matrices () and minimally -semipositive matrices () are two important subclasses of -semipositive matrices. In this paper, we establish the existence of bases for the real vector space of matrices, consisting of -positive matrices and of minimally -semipositive matrices. Sufficient conditions are determined for -semipositivity, in terms of the length of rows(columns) of the matrices. Furthermore, we discuss properties of -semipositive matrices involving product of matrices. At last, -semipositive matrices are described via entries of the matrices and equivalent -semipositive matrices are studied
