Related papers: Explicit Description of Centralizers for a Matrix

We introduce the n-th Heron variety as the realization space of the (squared) volumes of faces of an n-simplex. Our primary goal is to understand the extent to which Heron's formula, which expresses the area of a triangle as a function of…

We give a natural definition for the transitivity of a matrix. Using an endomorphism d of a base ring R and a transitive nxn matrix over the center Z(R), we construct the subalgebra M_{n}(R,d,T) of the full nxn matrix algebra M_{n}(R)…

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. We determine the maximal dimension for an affine subspace of $n$ by $n$ symmetric (or alternating) matrices with entries in $\mathbb{K}$ and with…

Explicit expressions are given for the actions and radial matrix elements of basic radial observables on multi-dimensional spaces in a continuous sequence of orthonormal bases for unitary SU(1,1) irreps. Explicit expressions are also given…

This papers studies centralizers of an element, $a$, in the nucleus of a non-associative algebra with a special type of valuation. We prove that the centralizer of $a$ is a free module of finite rank over the algebra generated by $a$.

A linear mapping $\phi$ on an algebra $\mathcal{A}$ is called a centralizable mapping at $G\in\mathcal{A}$ if $\phi(AB)=\phi(A)B=A\phi(B)$ for each $A$ and $B$ in $\mathcal{A}$ with $AB=G$, and $\phi$ is called a derivable mapping at…

Matrix polynomials given in an orthogonal basis are considered. Following the ideas of Mackey et al. "Vector spaces of Linearizations for Matrix Polynomials" (2006), the vec- tor spaces, called M1(P), M2(P) and DM(P), of potential…

We provide a general formula for Mueger's centralizer of any fusion subcategory of a braided fusion category containing a tannakian subcategory. This entails a description for Mueger's centralizer of all fusion subcategories of a group…

In the Determinant Maximization problem, given an $n\times n$ positive semi-definite matrix $\bf{A}$ in $\mathbb{Q}^{n\times n}$ and an integer $k$, we are required to find a $k\times k$ principal submatrix of $\bf{A}$ having the maximum…

An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal…

We present an optimization procedure for a seminal class of positive maps $\tau_{n,k}$ in the algebra of $n \times n$ complex matrices introduced and studied by Tanahasi and Tomiyama, Ando, Nakamura and Osaka. Recently, these maps were…

We consider a large class of matrix problems, which includes the problem of classifying arbitrary systems of linear mappings. For every matrix problem from this class, we construct Belitskii's algorithm for reducing a matrix to a canonical…

We study the problem of determining a matrix whose $k$th multiplicative compound is a prescribed matrix~$M$. The cardinality of the set of matrices whose $k$th multiplicative compound equals~$M$ is characterized in terms of $\rank(M)$. On…

This work is devoted to computing the centralizer $Z (L)$ of an ordinary differential operator (ODO) in the ring of differential operators. Non-trivial centralizers are known to be coordinate rings of spectral curves and contain the ring of…

We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the…

Let $n,k$ be fixed natural numbers with $1\le k\le n$ and let $A_{n+1,k,2k,\dots,sk}$ denote an $(n+1)\times (n+1)$ complex multidiagonal matrix having $s=[n/k]$ sub- and superdiagonals at distances $k,2k,\dots,sk$ from the main diagonal.…

The $k$-Means clustering problem on $n$ points is NP-Hard for any dimension $d\ge 2$, however, for the 1D case there exists exact polynomial time algorithms. Previous literature reported an $O(kn^2)$ time dynamic programming algorithm that…

In this work, we find base and dimension of a subspace appeared in works by V. A. Sharafutdinov. The same problem, but expressed in terms of polynomials from matrix minors, was initially solved by W. Hodge. The new result of this paper is…

Matrix multiplication is a fundamental task in almost all computational fields, including machine learning and optimization, computer graphics, signal processing, and graph algorithms (static and dynamic). Twin-width is a natural complexity…

Let $A$ be an abelian variety over a finite field $k$ with $|k|=q=p^m$. Let $\pi\in \text{End}_k(A)$ denote the Frobenius and let $v=\frac{q}{\pi}$ denote Verschiebung. Suppose the Weil $q$-polynomial of $A$ is irreducible. When…