Related papers: Determinantal inequalities for block triangular ma…

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal…

For any primitive matrix $M\in\mathbb{R}^{n\times n}$ with positive diagonal entries, we prove the existence and uniqueness of a positive vector $\mathbf{x}=(x_1,\dots,x_n)^t$ such that $M\mathbf{x}=(\frac{1}{x_1},\dots,\frac{1}{x_n})^t$.…

We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay…

Let G be a block matrix function with one diagonal block A being positive definite and the off diagonal blocks complex conjugates of each other. Conditions are obtained for G to be factorable (in particular, with zero partial indices) in…

In this paper, we investigate the determinants involving some trigonometric functions. We establish a connection between these determinants and the special values of Dirichlet L-functions, thereby extending Guo's results to arbitrary…

In this paper we study some determinant inequalities and matrix inequalities which have a geometrical flavour. We first examine some inequalities which place work of Macbeath [13] in a more general setting and also relate to recent work of…

We compute the determinant of $\sum_{n=1}^{N} \vec{A}^{(n)} \otimes \vec{B}^{(n)}$, where $\vec{A}^{(n)}$ is square and ${\vec{B}^{(n)}=\vec{x}^{(n)}{\vec{y}^{(n)}}^T}$ where $\vec{x}^{(n)}$ and $\vec{y}^{(n)}$ have length $N$.

Let $T$ be a tree with vertex set $\{1, \ldots, n\}$ such that each edge is assigned a nonzero weight. The squared distance matrix of $T,$ denoted by $\Delta,$ is the $n \times n$ matrix with $(i,j)$-element $d(i,j)^2,$ where $d(i,j)$ is…

Let $A, B$ and $X$ be $n\times n$ matrices such that $A, B$ are positive semidefinite. We present some refinements of the matrix Cauchy-Schwarz inequality by using some integration techniques and various refinements of the Hermite--Hadamard…

We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z. Consequently, results of Eilers,…

Let $A$ be an $n\times n$ real Toeplitz matrix satisfying $A+A^{\top}=2\mathbb J_n$, where $\mathbb J_n$ is the all-ones matrix.If $A_r(i,j)$ denotes the $r\times r$ contiguous submatrix of $A$ consisting of rows $i,i+1,\dots,i+r-1$ and…

In this article, we consider birational equivalence of exponential matrices. In characteristic zero, we give a birational classification of exponential matrices of size $n$-by-$n$ $(n \geq 2)$, which consists of two types. And in positive…

The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the…

We show that certain determinantal functions of multiple matrices, when summed over the symmetries of the cube, decompose into functions of the original matrices. These are shown to be true in complete generality; that is, no properties of…

Baur and Marsh computed the determinant of a matrix assembled from the cluster variables in a cluster algebra of type A. In this article we wish to describe two variations. On the one hand, we compute determinants of matrices assembled from…

The aim of this note is to give two new conceptual proofs of Ionescu-Weitzenb\"ock's inequality. The first one, which is a vector proof, provides us a geometric interpretation of the difference between the two sides of this inequality and…

We establish a bijection between the set of finite topological $T_0$-spaces (or partially ordered sets) and equivalence classes of square matrices. The absolute value of the determinant or the rank of these matrices serve as simple homotopy…

We prove that for writing the 3 by 3 permanent polynomial as a determinant of a matrix consisting only of zeros, ones, and variables as entries, a 7 by 7 matrix is required. Our proof is computer based and uses the enumeration of bipartite…

For a 4th order 3-dimensional symmetric tensor with its some entries $1$ or $-1$, we show the analytic sufficient and necessary conditions of its positive definiteness. By applying these conclusions, several strict inequalities is bulit for…

In this work the Erdos-Mordell's inequality is examined for the case of a triangle $ABC$ in the taxicab plane geometry. It is shown that the Erdos-Mordell's inequality $R_A + R_B + R_C \, \geq \, w \, (r_a + r_b + r_c)$ holds for triangles…