Related papers: Corners of normal matrices
Consider two symmetric $3 \times 3$ matrices $A$ and $B$ with entries in $GF(q)$, for $q=p^n$, $p$ an odd prime. The zero sets of $v^T Av$ and $v^T Bv$ can be viewed as (possibly degenerate) conics in the finite projective coordinate plane…
Let $A$ and $B$ be two -non necessarily bounded- normal operators. We give new conditions making their product normal. We also generalize a result by Deutsch et al on normal products of matrices.
Matrices over the dual numbers are considered. We propose an approach to classify these matrices up to similarity. Some preliminary results on the realization of this approach are obtained. In particular, we produce explicitly canonical…
Condition numbers of random polynomial systems have been widely studied in the literature under certain coefficient ensembles of invariant type. In this note we introduce a method that allows us to study these numbers for a broad family of…
We survey partial geometric designs and investigate their concurrences of points. The concurrence matrix of a design, which encodes the concurrences of pairs of points, can be used in the classification of designs in some extent. An…
We provide new examples of integrable rational maps in four dimensions with two rational invariants, which have unexpected geometric properties, as for example orbits confined to non algebraic varieties, and fall outside classes studied by…
An n \times n matrix A has a normal defect of k if there exists an (n+k) \times (n+k) normal matrix A_{ext} with A as a leading principal submatrix and k minimal. In this paper we compute the normal defect of a special class of 4 \times 4…
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…
We combine the total variation flow suitable for crystal modeling and image analysis with the dynamic boundary conditions. We analyze the behavior of facets at the parts of the boundary where these conditions are imposed. We devote…
In this paper we consider planar polygons with parallel opposite sides. This type of polygons can be regarded as discretizations of closed convex planar curves by taking tangent lines at samples with pairwise parallel tangents. For this…
We study generalization of median triangles on the plane with two complex parameters. By specialization of the parameters, we produce periodical motion of a triangle whose vertices trace each other on a common closed orbit.
We prove two inequalities regarding the ratio $\det(A+D)/\det A$ of the determinant of a positive-definite matrix $A$ and the determinant of its perturbation $A+D$. In the first problem, we study the perturbations that happen when positive…
The ensemble of antagonistic matrices is introduced and studied. In antagonistic matrices the entries $\mathcal A_{i,j}$ and $\mathcal A_{j,i}$ are real and have opposite signs, or are both zero, and the diagonal is zero. This…
We investigate two-sided bounds for operator norms of random matrices with unhomogenous independent entries. We formulate a lower bound for Rademacher matrices and conjecture that it may be reversed up to a universal constant. We show that…
Let $\Omega_n$ denote the class of $n \times n$ doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in $\Omega_n$. The main question is: which $A \in…
We analyze surface patches with a corner that is rounded in the sense that the partial derivatives at that point are antiparallel. Sufficient conditions for $G^1$ smoothness are given, which, up to a certain degenerate case, are also…
In this work, we present a cogredient standard form of an orthogonal space over a finite local ring of odd characteristic.
We collect here elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange \& Hermite…
Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics…
We discuss several conjectures about the real-rootedness of polynomials whose coefficients are determinants of coefficients of a real-rooted polynomial. We also consider some questions about matrices generalizing totally positive matrices,…