Related papers: Sums of two triangularizable quadratic matrices ov…

A commutator of unipotent matrices of index 2 is a matrix of the form $XYX^{-1}Y^{-1}$, where $X$ and $Y$ are unipotent matrices of index 2, that is, $X\ne I_n$, $Y\ne I_n$, and $(X-I_n)^2=(Y-I_n)^2=0_n$. If $n>2$ and $\mathbb F$ is a field…

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $n\times n$ matrix ring, so $R\cong M_{n}(S)$ for some ring $S$, if and only if it contains a…

Let $R$ be an associative ring with identity $1$. We describe all matrices in $T_n(R)$ the ring of $n\times n$ upper triangular matrices over $R$ ($n\in \mathbb{N}$), and $T_{\infty}(R)$ the ring of infinite upper triangular matrices over…

We study entrywise scalar quantization of two matrices prior to multiplication. Given $A\in R^{m\times k}$ and $B\in R^{k\times n}$, we quantize entries of $A$ and $B$ independently using scalar quantizers with $K_X$ and $K_Y$ levels per…

We estimate the truncated double trigonometric series $\sum_{n=0}^{N}\sum_{m=0}^{M}a_{mn} {e}^{2\pi \imath \left(m x+n y\right)}$, $a_{mn} \in\mathbb{C}$, in Lebesgue spaces with mixed norms in terms of the $p^{th}-q^{th}$ power finite…

The Matrix Waring problem is if we can write every matrix as a sum of $k$-th powers. Here, we look at the same problem for triangular matrix algebra $T_n(\mathbb{F}_q)$ consisting of upper triangular matrices over a finite field. We prove…

We evidence a family $\mathcal{X}$ of square matrices over a field $\mathbb{K}$, whose elements will be called X-matrices. We show that this family is shape invariant under multiplication as well as transposition. We show that $\mathcal{X}$…

Sums-of-squares formulas over the integers have been studied extensively using their equivalence to consistently signed intercalate matrices. This representation, combined with combinatorial arguments, has been used to produce…

This paper delves into the equivalence problem of Smith forms for multivariate polynomial matrices. Generally speaking, multivariate ($n \geq 2$) polynomial matrices and their Smith forms may not be equivalent. However, under certain…

Let $K$ be an algebraically closed field of characteristic $0$ and let $M_n(K)$, $n \ge 3$, be the matrix ring over $K$. We will show that the image of any multilinear polynomial in four variables evaluated on $M_n(K)$ contains all matrices…

In this article we obtain a general polynomial identity in $k$ variables, where $k\geq 2$ is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a $k \times k$ matrix.…

By means of $q$-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus $x^2+y^2$ for some integers $x$ and $y$ with $x\not\equiv y…

We use the spectral theory of Hilbert-Maass forms for real quadratic fields to obtain the asymptotics of some sums involving the number of representations as a sum of two squares in the ring of integers.

The iterative method of Sinkhorn allows, starting from an arbitrary real matrix with non-negative entries, to find a so-called 'scaled matrix' which is doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and with all…

A real symmetric matrix $M$ is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size $d$. The smallest such $d$ is called the (complex) completely positive semidefinite…

We construct the Generalized Monodromy matrix $\mathcal{\hat{M}}(\omega)$ of two dimensional string effective action by introducing the T-duality group properties.The integrability conditions with general solutions depending on spectral…

We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational…

Let K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K with finite positive dimension and let A,B denote a tridiagonal pair on V. We make an assumption about this pair. Let q denote a nonzero…

This paper investigates equivalence of square multivariate polynomial matrices with the determinant being some power of a univariate irreducible polynomial. We first generalized a global-local theorem of Vaserstein. Then we proved these…

We present a necessary and sufficient condition for the separability of multipartite quantum states, this criterion also tells us how to write a multipartite separable state as a convex sum of separable pure states. To work out this…