Related papers: A note on real similarity to a diagonal dominant m…
We continue the study of real polynomials acting entrywise on matrices of fixed dimension to preserve positive semidefiniteness, together with the related analysis of order properties of Schur polynomials. Previous work has shown that,…
In this paper we describe some properties of companion matrices and demonstrate some special patterns that arise when a Toeplitz or a Hankel matrix is multiplied by a related companion matrix. We present a new condition, generalizing known…
A real square matrix is Perron-like if it has a real eigenvalue $s$, called the principal eigenvalue of the matrix, and $\mbox{Re}\,\mu<s$ for any other eigenvalue $\mu$. Nonnegative matrices and symmetric ones are typical examples of this…
The concepts of differentiation and integration for matrices were introduced for studying zeros and critical points of complex polynomials. Any matrix is differentiable, however not all matrices are integrable. The purpose of this paper is…
Every matrix polynomial $\mathbf{f}_n$ can be written in the form \[ \mathbf{f}_n(z)=\mathbf{h}(z^2)+z\,\mathbf{g}_n(z^2). \] The matrix polynomial $\mathbf{f}_{2m}$ is said to be of Hurwitz type if the expression…
This paper is concentrated on the classification of permutation matrix with the permutation similarity relation, mainly about the canonical form of a permutational similar equivalence class, the cycle matrix decomposition of a permutation…
We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the…
Toeplitz matrices are characterized by their constant diagonals, have been extensively studied in various settings, including over real and complex numbers. However, their study over quaternions is quite sparse. In this paper, we…
Two matrices are said to be principal minor equivalent if they have equal corresponding principal minors of all orders. We give a characterization of principal minor equivalence and a deterministic polynomial time algorithm to check if two…
The symmetry properties of the dynamical matrix are well described in multiple classic textbooks. This short paper revisits the issue to demonstrate alternative form of dynamical matrix which explicitly shows its symmetry and reality in…
New Frobenius structures on Hurwitz spaces are found. A Hurwitz space is considered as a real manifold; therefore the number of coordinates is twice as large as the number of coordinates on Hurwitzs Frobenius manifolds of Dubrovin. Simple…
In these notes we explore a variety of models comprising a large number of constituents. An emphasis is placed on integrals over large Hermitian matrices, as well as quantum mechanical models whose degrees of freedom are organised in a…
We develop a procedure for determining whether a square complex matrix is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. Our approach has several advantages over existing methods. We discuss these differences and…
We discuss several conjectures about the real-rootedness of polynomials whose coefficients are determinants of coefficients of a real-rooted polynomial. We also consider some questions about matrices generalizing totally positive matrices,…
We prove conditions for equality between the extreme eigenvalues of a matrix and its quotient. In particular, we give a lower bound on the largest singular value of a matrix and generalize a result of Finck and Grohmann about the largest…
The task of analytically diagonalizing a tridiagonal matrix can be considerably simplified when a part of the matrix is uniform. Such quasi-uniform matrices occur in several physical contexts, both classical and quantum, where…
This paper elaborates on a relationship between matrix-valued Herglotz-Nevanlinna functions and Hurwitz stable matrix polynomials, which generalizes the corresponding classical stability criterion. The main motivation comes from the…
Due to their rich algebraic structures and various applications, circulant matrices have been of interest and continuously studied. In this paper, the notions of Binomial-related matrices have been introduced. Such matrices are circulant…
The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we…
In this paper we give conditions on a matrix which guarantee that it is similar to a centrosymmetric matrix. We use this conditions to show that some $4 \times 4$ and $6 \times 6$ Toeplitz matrices are similar to centrosymmetric matrices.…