In this paper we will look at the well known interlacing problem, but here we consider the result for Hermitian matrices in the Minkowski space, an indefinite inner product space with one negative square. More specific, we consider the n×n matrix A=[Ju−u⁎a] with a∈R, J=J⁎ and u∈Cn−1. Then A is H-selfadjoint with respect to the matrix H=In−1⊕(−1). The canonical form for the pair (A,H) plays an important role and the sign characteristic coupled to the pair is also discussed.

The nonnegative inverse eigenvalue problem is considered in this paper with the additional restriction of fixed zero patterns in the matrix. A full analysis of the $3\times 3$ case is given. Some remarks on the four-dimensional case are made.

In this paper small rank perturbations of H-expansive and H-unitary matrices are explored. Particular attention is given to the location of eigenvalues with respect to the unit circle for these classes of matrices. The canonical form (in the H-unitary case) and the simple form (in the H-expansive case) for the pair $$(A,H)$$ will be the starting point.

In this essay algebraic Riccati equations will be discussed. It turns out that Hermitian solutions of algebraic Riccati equations which originate from systems and control theory may be studied in terms of invariant Lagrangian subspaces of matrices which are selfadjoint in an indefinite inner product. The essay will describe briefly certain problems in systems and control theory where the algebraic Riccati equation plays a role. The focus in the main part of the essay will be on those aspects of the theory of matrices in indefinite inner product spaces that were motivated and largely influenced by the connection with the study of Hermitian solutions of algebraic Riccati equations. This includes the description of uniqueness and stability of invariant Lagrangian subspaces and of invariant maximal semidefinite subspaces of matrices that are selfadjoint in the indefinite inner product, which leads to the concept of the sign condition. Also, it is described how the inertia of solutions of a special type of algebraic Riccati equation may be described completely in terms of the invariant Lagrangian subspaces connected with the solutions.

Inspired by the paper of Groenewald, Kaashoek and Ran (2017) [19], we present an operator-theoretic approach to provide further insight and simpler computational formulas for the Wiener-Hopf indices of a rational matrix valued function taking unimodular values on the unit circle.

This paper is concerned with the Wiener-Hopf indices of unitary-valued rational matrix functions on the imaginary axis. These indices play a role in the Fredholm theory for Wiener-Hopf integral operators. Our main result gives formulas for the Wiener-Hopf indices in terms of the matrices appearing in realizations of the factors in a Douglas-Shapiro-Shields factorization of the unitary-valued function. Two approaches to this problem are presented: one direct approach using operator theoretic methods, and a second approach using the Cayley transform which allows to use results for an analogous problem regarding unitary-valued functions on the unit circle and corresponding Toeplitz operators.

We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA, 258:187-194, 1997. Let $p(\lambda)$ be any polynomial over $\mathbb{Q}$, and let $A\in M_n(\mathbb{Q})$ have irreducible characteristic polynomial $f(\lambda)$ with degree $n$. We provide necessary and sufficient conditions for the existence of a solution $X\in M_n(\mathbb{Q})$ of the polynomial matrix equation $p(X) = A.$ Specifically, we find necessary and sufficient conditions for $f(p(\lambda))$ to have a factor of degree $n$ over $\mathbb{Q}.$

We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA 258 (1997), 187-194. Let p(λ) be any polynomial over ℚ and let A∈ M_n(ℚ) have irreducible characteristic polynomial f(λ) with degree n. We provide necessary and sufficient conditions for the existence of a solution X∈ M_n(ℚ) of the polynomial matrix equation p(X) = A. Specifically, we find necessary and sufficient conditions for f(p(λ)) to have a factor of degree n over ℚ.

The synchronization of power generators is an important condition for the proper functioning of a power system, in which the fluctuations in frequency and the phase angle differences between the generators are sufficiently small when subjected to stochastic disturbances. Serious fluctuations can prompt desynchronization, which may lead to widespread power outages. Here, we model the stochastic disturbance by a Brownian motion process in the linearized system of the non-linear power systems and characterize the fluctuations by the variances of the frequency and the phase angle differences in the invariant probability distribution. We propose a method to calculate the variances of the frequency and the phase angle differences. For the system with uniform disturbance-damping ratio, we derive explicit formulas for the variance matrices of the frequency and the phase angle differences. It is shown that the fluctuation of the frequency at a node depends on the disturbance-damping ratio and the inertia at this node only, and the fluctuations of the phase angle differences in the lines are independent of the inertia. In particular, the synchronization stability is related to the cycle space of the network. We reveal the influences of constructing new lines and increasing capacities of lines on the fluctuations in the phase angle differences in the existing lines. The results are illustrated for the transmission system of Shandong Province of China. For the system with non-uniform disturbance-damping ratio, we further obtain bounds of the variance matrices.

Let H be a field with Q subset of H subset of C, and let p(lambda) be a polynomial in H[lambda], and let A is an element of Hnxn be nonderogatory. In this paper we consider the problem of finding a solution X is an element of Hnxn to p(X) = A. A necessary condition for this to be possible is already known from a paper by M.P. Drazin: Exact rational solutions of the matrix equation A = p(X) by linearization. Under an additional condition we provide an explicit construction of such solutions. The similarities and differences with the derogatory case will be discussed as well. One of the tools needed in the paper is a new canonical form, which may be of independent interest. It combines elements of the rational canonical form with elements of the Jordan canonical form.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).

As in the paper Groenewald et al. (2017) our aim is to obtain explicitly the Wiener-Hopf indices of a rational m x m matrix function R(z) that has no poles and no zeros on the unit circle T but, in contrast with Groenewald et al. (2017), the function R(z) is not required to be unitary on the unit circle. On the other hand, using a Douglas-Shapiro-Shields type of factorization, we show that R(z) factors as R(z) = E (z)W(z), where E (z) and W(z) are rational m x m matrix functions, E (z) is unitary on the unit circle and W(z) is an invertible outer function. Furthermore, the fact that E (z) is unitary on the unit circle allows us to factor as E(z) = V(z)W*(z) where V (z) and W (z) are rational bi-inner m xm matrix functions. The latter allows us to solve the Wiener-Hopf indices problem. To derive explicit formulas for the functions V (z) and W (z) requires additional realization properties of the function E(z) which are given in the last two sections. (c) 2022 The Author(s). Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

A conjecture from a paper by J. Bierkens and A.C.M. Ran concerning the location of eigenvalues of rank one perturbations of singular M-matrices is shown to be false in dimension four and higher, but true for dimension two, as well as for dimension three with an additional condition on the perturbation.

We present in this note a correction to Theorem 17 in Ran and Wojtylak (Compl. Anal. Oper. Theory 15:44, 2021) and sharpen the estimates for eigenvalues of parametric rank one perturbations given in that theorem.

A result on the structure of expansive matrices in an indefinite inner product space is derived, which exhibits the largest unitary compression of the matrix.

The complex matrix representation for a quaternion matrix is used in this paper to find necessary and sufficient conditions for the existence of an H-selfadjoint mth root of a given H-selfadjoint quaternion matrix. In the process, when such an H-selfadjoint mth root exists, its construction is also given.

Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an $H$-polar decomposition are found. In the process, an equivalent to Witt's theorem on extending $H$-isometries to $H$-unitary matrices is given for quaternion matrices.