Related papers: On minimality of determinantal varieties

We study the differential and metric structures of the set of real square roots of a non-singular real matrix, under the assumption that the matrix and its square roots are semi-simple, or symmetric, or orthogonal.

We show that the shortest nonzero polynomials vanishing on bounded-rank matrices and skew-symmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that in the ideal generated by all $t$-minors…

We show that the set of $m \times m$ complex skew-symmetric matrix polynomials of even grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$ is the closure of the single set of matrix polynomials with certain, explicitly…

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…

Some elementary considerations are presented concerning Catenoids and their stability, separable minimal hypersurfaces, minimal surfaces obtainable by rotating shapes, determinantal varieties, minimal tori in S3, the minimality in Rnk of…

A vector space A of matrices is called rank-critical if any vector space that properly contains A has a strictly higher generic rank. I present a sufficient condition for A to be rank-critical, and apply this condition to prove that certain…

We consider the problem of finding the smallest rank of a complex matrix whose absolute values of the entries are given. We call this minimum the phaseless rank of the matrix of the entrywise absolute values. In this paper we study this…

In this paper we express the eigenvalues of a sort of real heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From these prescribed eigenvalues we compute also…

Let M be a p-by-q matrix with nonnegative entries. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices $A_i, B_j$ of size $k \times k$ such that $M_{ij} =…

We study a collection of Hessenberg varieties in the type A flag variety associated to a nonzero semisimple matrix whose conjugacy class has minimal dimension. We prove each such minimal semisimple Hessenberg variety is a union Richardson…

We give one more proof of the fact that symplectic matrices over real and complex fields have determinant one. While this has already been proved many times, there has been lasting interest in finding an elementary proof. Our result is…

We prove minimal entropy rigidity for complete, finite volume manifolds locally isometric to a product of rank one symmetric spaces of dimension at least 3: the locally symmetric metric uniquely minimizes (normalized) entropy among all…

We use representation theory to construct spaces of matrices of constant rank. These spaces are parametrized by the natural representation of the general linear group or the symplectic group. We present variants of this idea, with more…

Symplectic eigenvalues are conventionally defined for symmetric positive-definite matrices via Williamson's diagonal form. Many properties of standard eigenvalues, including the trace minimization theorem, are extended to the case of…

In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special…

We prove that the local components of an automorphic representation of an adelic semisimple group have equal rank in the sense defined earlier by the second author. Our theorem is an analogue of the results previously obtained by Howe, Li,…

The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are…

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically,…

In this paper, by studying the position of umbilical normal vectors in the normal bundle, we prove that pseudo-umbilical totally real submanifolds with flat normal connection in non-flat complex space forms must be minimal.

In this paper, using Descartes law of sign, we provide elementary proof of results on the number of real eigenvalues of real matrices of which certain properties on the signs of their principal minors are known. More precisely, we analyze…