Related papers: Spaces of matrices with few eigenvalues (II)

If $f$ is an endomorphism of a finite dimensional vector space over a field $K$ then an invariant subspace $X \subseteq V$ is called hyperinvariant (respectively, characteristic) if $X$ is invariant under all endomorphisms (respectively,…

Given an arbitrary (commutative) field K, let V be a linear subspace of M_n(K) consisting of matrices of rank lesser than or equal to some r<n. A theorem of Atkinson and Lloyd states that, if dim V>nr-r+1 and #K>r, then either all the…

It is well known that the set of all $ n \times n $ matrices with distinct eigenvalues is a dense subset of the set of all real or complex $ n \times n $ matrices. In [Hartfiel, D. J. Dense sets of diagonalizable matrices. Proc. Amer. Math.…

A minimal homogeneous generating system of the algebra of semi-invariants of tuples of two-by-two matrices over an infinite field of characteristic two or over the ring of integers is given. In an alternative interpretation this yields a…

We show that self-dual 2-forms in 2n dimensional spaces determine a $n^2-n+1$ dimensional manifold ${\cal S}_{2n}$ and the dimension of the maximal linear subspaces of ${\cal S}_{2n}$ is equal to the (Radon-Hurwitz) number of linearly…

Let K be a field and let M_n(K) denote the space of n x n matrices with entries in K. Let M be a subspace of M_n(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the…

We consider the set $\mathcal{M}_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb Z; H)$ with a given characteristic…

Let $F$ be an infinite field, and let $M_{n}(F)$ be the algebra of $n\times n$ matrices over $F$. Suppose that this algebra is equipped with an elementary grading whose neutral component coincides with the main diagonal. In this paper, we…

In applications of linear algebra including nuclear physics and structural dynamics, there is a need to deal with uncertainty in the matrices. We focus on matrices that depend on a set of parameters $\omega$ and we are interested in the…

Let $\mathbb{D}$ be a division ring and $\mathbb{F}$ be a subfield of the center of $\mathbb{D}$ over which $\mathbb{D}$ has finite dimension $d$. Let $n,p,r$ be positive integers and $\mathcal{V}$ be an affine subspace of the…

Given an integral domain A we consider the set of all integral elements over A that can occur as an eigenvalue of a symmetric matrix over A. We give a sufficient criterion for being such an element. In the case where A is the ring of…

A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair $(A, C)$ is introduced in this paper. The 2DEVP can be viewed as a linear algebraic formulation of the well-known eigenvalue optimization problem of the parameter…

Two subspaces of a vector space are here called ``nonintersecting'' if they meet only in the zero vector. The following problem arises in the design of noncoherent multiple-antenna communications systems. How many pairwise nonintersecting…

We describe properties of a Hermitian square matrix M in M_n(C) equivalent to that of having minimal quotient norm in the following sense: ||M|| <= ||M+D|| for all real diagonal matrices D in M_n(C) and || || the operator norm. These…

In this article, we address the problem of determining a domain in $\mathbb{R}^N$ that minimizes the first eigenvalue of the Lam\'e system under a volume constraint. We begin by establishing the existence of such an optimal domain within…

Let V be a linear subspace of M_{n,p}(K) with codimension lesser than n, where K is an arbitrary field and n >=p. In a recent work of the author, it was proven that V is always spanned by its rank p matrices unless n=p=2 and K is isomorphic…

We classify all Mathieu subspaces of ${\rm Mat}_n(K)$ of codimension less than $n$, under the assumption that ${\rm char\,} K = 0$ or ${\rm char\,} K \ge n$. More precisely, we show that any proper Mathieu subspace of ${\rm Mat}_n(K)$ of…

Let $S_{n}$ denote the space of all $n \times n$ real symmetric matrices. For n=2 or n>2 we characterize maps F from $S_{n}$ to $S_{m}$ which preserve adjacency, i.e. if rank(A-B)=1, then rank(F(A)-F(B))=1.

A minimal (by inclusion) generating set for the algebra of semi-invariants of a quiver of dimension (2,...,2) is established over an infinite field of arbitrary characteristic. The mentioned generating set consists of the determinants of…

We consider the problem of finding the best nonnegative rank-2 approximation of an arbitrary nonnegative matrix. We first revisit the theory, including an explicit parametrization of all possible nonnegative factorizations of a nonnegative…