Related papers: Explicit diagonalization of an anti-triangular Ces…
Trigonometric formulas for eigenvalues of $3 \times 3$ matrices that build on Cardano's and Vi\`ete's work on algebraic solutions of the cubic are numerically unstable for matrices with repeated eigenvalues. This work presents numerically…
Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to…
We prove the decomposition of arbitrary diagonal operators into tensor and matrix products of smaller matrices, focusing on the analytic structure of the resulting formulas and their inherent symmetries. Diagrammatic representations are…
This paper explicitly details the relation between $M$-matrices, nonnegative roots of nonnegative matrices, and the embedding problem for finite-state stationary Markov chains. The set of nonsingular nonnegative matrices with arbitrary…
This note discusses an interesting matrix factorization called the CUR Decomposition. We illustrate various viewpoints of this method by comparing and contrasting them in different situations. Additionally, we offer a new characterization…
In this paper, we introduce two new forms of the dual Hartwig-Spindelb{\"o}ck decomposition and employ them to derive explicit representations for several classes of dual generalized inverses. Building on these representations, we further…
The paper introduces a method of partial fractions with matrix coefficients and its applications to finding chains of generalized eigenvectors, to evaluation of matrix exponentials, and to solution of linear systems of ordinary differential…
This note presents a summary and review of various conditions and characterizations for matrix stability (in particular diagonal matrix stability) and matrix stabilizability.
We associate a complete intersection singularity to a graded matrix factorization of size two of a polynomial in three variables. We show that we get an inverse to the reduction of singularities considered by C.T.C.Wall. We study this for…
Block tridiagonal matrices arise in applied mathematics, physics, and signal processing. Many applications require knowledge of eigenvalues and eigenvectors of block tridiagonal matrices, which can be prohibitively expensive for large…
The multidimensional chain rule formula for analytic functions and its generalisation to higher derivatives perfectly work in the algebraic setting in characteristic zero. In positive characteristic one runs into problems due to…
For the eigenvalues of $p$ complex hermitian $n\times n$ matrices coupled in a chain, we give a method of calculating the spacing functions. This is a generalization of the one matrix case which has been known for a long time.
The triangulations of a regular convex polygon are enumerated according to the number of diagonals parallel to a fixed edge. The enumeration uses the Shapiro convolution identity, as well as an interpretation of this identity in terms of…
We find the spectrum and eigenvectors of an arbitrary irreducible complex tridiagonal matrix with two-periodic main diagonal provided that the spectrum and eigenvectors of the matrix with the same sub- and superdiagonals and zero main…
We prove that the following statements are equivalent: a linear matrix equation with parameters forming a commuting set of diagonalizable matrices is consistent, a certain matrix constructed with the Drazin inverse is a solution of this…
The spectral properties of a class of band matrices are investigated. The reconstruction of matrices of this special class from given spectral data is also studied. Necessary and sufficient conditions for that reconstruction are found. The…
In this paper, we generalize the Markov triples in two different directions. One is generalization in direction of using the $q$-deformation of rational number introduced by \cite{MO} in connection with cluster algebras, quantum topology…
We introduce right eigenvalues and subeigenvalues for square dual complex matrices. An $n \times n$ dual complex Hermitian matrix has exactly $n$ right eigenvalues and subeigenvalues, which are all real. The Hermitian matrix is positive…
A classical problem for Markov chains is determining their stationary (or steady-state) distribution. This problem has an equally classical solution based on eigenvectors and linear equation systems. However, this approach does not scale to…
This article settles the convergence question for multivariate barycentric subdivision schemes with nonnegative masks on complete metric spaces of nonpositive Alexandrov curvature, also known as Hadamard spaces. We establish a link between…