arXiv subjects

Andrii Dmytryshyn

Publications and source records attributed to Andrii Dmytryshyn.

18 recordsLinked to original sources

arXiv2026

Mixed-precision algorithms for solving the Sylvester matrix equation

We consider the solution of the Sylvester equation $AX+XB=C$ in mixed precision. We derive a new iterative refinement scheme to solve perturbed quasi-triangular Sylvester equations; our rounding error analysis provides sufficient conditions for convergence and a bound on the attainable relative residual. We leverage this iterative scheme to solve the general Sylvester equation. The new algorithms compute the Schur decomposition of the coefficient matrices $A$ and $B$ in lower than working precision, use the low-precision Schur factors to obtain an approximate solution to the perturbed quasi-triangular equation, and iteratively refine it to obtain a working-precision solution. In order to solve the original equation to working precision, the unitary Schur factors of the coefficient matrices must be unitary to working precision, but this is not the case if the Schur decomposition is computed in low precision. We propose two effective approaches to address this: one is based on re-orthonormalization in working precision, and the other on explicit inversion of the almost-unitary factors. The two mixed-precision algorithms thus obtained are tested on various Sylvester and Lyapunov equations from the literature. Our numerical experiments show that, for both types of equations, the new algorithms are at least as accurate as existing ones. Our cost analysis, on the other hand, suggests that they would typically be faster than mono-precision alternatives if implemented on hardware that natively supports low precision.

math.NA↗

arXiv2026

Distance to nearest skew-symmetric matrix polynomials of bounded rank

We propose an algorithm that approximates a given matrix polynomial of degree $d$ by another skew-symmetric matrix polynomial of a specified rank and degree at most $d$. The algorithm is built on recent advances in the theory of generic eigenstructures and factorizations for skew-symmetric matrix polynomials of bounded rank and degree. Taking into account that the rank of a skew-symmetric matrix polynomial is even, the algorithm works for any prescribed even rank greater than or equal to $2$ and produces a skew-symmetric matrix polynomial of that exact rank. We also adapt the algorithm for matrix pencils to achieve a better performance. Lastly, we present numerical experiments for testing our algorithms and for comparison to the previously known ones.

math.NA↗

arXiv2025

Minimal degenerations of orbits of skew-symmetric matrix pencils

Complete eigenstructure, e.g., eigenvalues with multiplicities and minimal indices, of a skew-symmetric matrix pencil may change drastically if the matrix coefficients of the pencil are subjected to (even small) perturbations. These changes can be investigated qualitatively by constructing the stratification (closure hierarchy) graphs of the congruence orbits of the pencils. The results of this paper facilitate the construction of such graphs by providing all closest neighbours for a given node in the graph. More precisely, we prove a necessary and sufficient condition for one congruence orbit of a skew-symmetric matrix pencil, A, to belong to the closure of the congruence orbit of another pencil, B, such that there is no pencil, C, whose orbit contains the closure of the orbit of A and is contained in the closure of the orbit of B.

math.RT↗

arXiv2025

Canonical forms for pairs of matrices associated with Lagrangian and Dirac subspaces

We derive the canonical forms for a pair of $n\times n$ complex matrices $(E,Q)$ under transformations $(E,Q) \rightarrow (UEV,U^{-T}QV)$, and $(E,Q) \rightarrow (UEV,U^{-*}QV)$, where $U$ and $V$ are nonsingular complex matrices. We, in particular, consider the special cases of $E^TQ$ and $E^*Q$ being (skew-)symmetric and (skew-)Hermitian, respectively, that are associated with Lagrangian and Dirac subspaces and related linear-time invariant dissipative Hamiltonian descriptor systems.

math.RT↗

arXiv2024

Minimal rank factorizations of polynomial matrices

We investigate rank revealing factorizations of $m \times n$ polynomial matrices $P(λ)$ into products of three, $P(λ) = L(λ) E(λ) R(λ)$, or two, $P(λ) = L(λ) R(λ)$, polynomial matrices. Among all possible factorizations of these types, we focus on those for which $L(λ)$ and/or $R(λ)$ is a minimal basis, since they have favorable properties from the point of view of data compression and allow us to relate easily the degree of $P(λ)$ with some degree properties of the factors. We call these factorizations minimal rank factorizations. Motivated by the well-known fact that, generically, rank deficient polynomial matrices over the complex field do not have eigenvalues, we pay particular attention to the properties of the minimal rank factorizations of polynomial matrices without eigenvalues. We carefully analyze the degree properties of generic minimal rank factorizations in the set of complex $m \times n$ polynomial matrices with normal rank at most $r< \min \{m,n\}$ and degree at most $d$, and we prove that there are only $rd+1$ different classes of generic factorizations according to the degree properties of the factors and that all of them are of the form $L(λ) R(λ)$, where the degrees of the $r$ columns of $L(λ)$ differ at most by one, the degrees of the $r$ rows of $R(λ)$ differ at most by one, and, for each $i=1, \ldots, r$, the sum of the degrees of the $i$th column of $L(λ)$ and of the $i$th row of $R(λ)$ is equal to $d$. Finally, we show how these sets of polynomial matrices with generic factorizations are related to the sets of polynomial matrices with generic eigenstructures.

math.NA↗

arXiv2023

Even grade generic skew-symmetric matrix polynomials with bounded rank

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 described, complete eigenstructure. This complete eigenstructure corresponds to the most generic $m \times m$ complex skew-symmetric matrix polynomials of even grade $d$ and rank at most $2r$. The analogous problem for the case of skew-symmetric matrix polynomials of odd grade is solved in [Linear Algebra Appl., 536:1-18, 2018].

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arXiv2023

Versal deformations: A tool of linear algebra

Versal deformation of a matrix A is a normal form to which all matrices A + E, close to A, can be reduced by similarity transformation smoothly depending on the entries of A + E. In this paper we discuss versal deformations and their use in codimension computations, in investigation of closure relations of orbits and bundles, in studying changes of canonical forms under perturbations, as well as in reduction of unstructured perturbations to structured perturbations.

math.RT↗

arXiv2023

Schur decomposition of several matrices

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it.

math.CO↗

arXiv2022

Generic eigenstructures of Hermitian pencils

We obtain the generic complete eigenstructures of complex Hermitian $n\times n$ matrix pencils with rank at most $r$ (with $r\leq n$). To do this, we prove that the set of such pencils is the union of a finite number of bundle closures, where each bundle is the set of complex Hermitian $n\times n$ pencils with the same complete eigenstructure (up to the specific values of the finite eigenvalues). We also obtain the explicit number of such bundles and their codimension. The cases $r=n$, corresponding to general Hermitian pencils, and $r<n$ exhibit surprising differences, since for $r<n$ the generic complete eigenstructures can contain only real eigenvalues, while for $r=n$ they can contain real and non-real eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures of Hermitian pencils.

math.NA↗

arXiv2020

Recovering a perturbation of a matrix polynomial from a perturbation of its linearization

A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory results for linearizations need to be related back to matrix polynomials. In this paper we present an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations.

math.NA↗

arXiv2019

Generic symmetric matrix polynomials with bounded rank and fixed odd grade

We determine the generic complete eigenstructures for $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. More precisely, we show that the set of $n \times n$ complex symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and rank at most $r$ is the union of the closures of the $\lfloor rd/2\rfloor+1$ sets of symmetric matrix polynomials having certain, explicitly described, complete eigenstructures. Then, we prove that these sets are open in the set of $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. In order to prove the previous results, we need to derive necessary and sufficient conditions for the existence of symmetric matrix polynomials with prescribed grade, rank, and complete eigenstructure, in the case where all their elementary divisors are different from each other and of degree $1$. An important remark on the results of this paper is that the generic eigenstructures identified in this work are completely different from the ones identified in previous works for unstructured and skew-symmetric matrix polynomials with bounded rank and fixed grade larger than one, because the symmetric ones include eigenvalues while the others not. This difference requires to use new techniques.

math.NA↗

arXiv2018

Generic symmetric matrix pencils with bounded rank

We show that the set of $n \times n$ complex symmetric matrix pencils of rank at most $r$ is the union of the closures of $\lfloor r/2\rfloor +1$ sets of matrix pencils with some, explicitly described, complete eigenstructures. As a consequence, these are the generic complete eigenstructures of $n \times n$ complex symmetric matrix pencils of rank at most $r$. We also show that these closures correspond to the irreducible components of the set of $n\times n$ symmetric matrix pencils with rank at most $r$ when considered as an algebraic set.

math.SP↗

arXiv2018

Miniversal deformations of pairs of symmetric matrices under congruence

For each pair of complex symmetric matrices $(A,B)$ we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices $(\widetilde{A},\widetilde{B})$, close to $(A,B)$ can be reduced by congruence transformation that smoothly depends on the entries of $\widetilde{A}$ and $\widetilde{B}$. Such a normal form is called a miniversal deformation of $(A,B)$ under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from $(A,B)$ to its miniversal deformation.

math.RT↗

arXiv2017

Generalization of Roth's solvability criteria to systems of matrix equations

W.E. Roth (1952) proved that the matrix equation $AX-XB=C$ has a solution if and only if the matrices $\left[\begin{matrix}A&C\\0&B\end{matrix}\right]$ and $\left[\begin{matrix}A&0\\0&B\end{matrix}\right]$ are similar. A. Dmytryshyn and B. Kågström (2015) extended Roth's criterion to systems of matrix equations $A_iX_{i'}M_i-N_iX_{i''}^{σ_i} B_i=C_i$ $(i=1,\dots,s)$ with unknown matrices $X_1,\dots,X_t$, in which every $X^σ$ is $X$, $X^T$, or $X^*$. We extend their criterion to systems of complex matrix equations that include the complex conjugation of unknown matrices. We also prove an analogous criterion for systems of quaternion matrix equations.

math.RT↗

arXiv2017

Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade

We show that the set of $m \times m$ complex skew-symmetric matrix polynomials of odd 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 the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic $m \times m$ complex skew-symmetric matrix polynomials of odd grade $d$ and rank at most $2r$. In particular, this result includes the case of skew-symmetric matrix pencils ($d=1$).

math.RA↗

arXiv2016

Generic matrix polynomials with fixed rank and fixed degree

The set ${\cal P}^{m\times n}_{r,d}$ of $m \times n$ complex matrix polynomials of grade $d$ and (normal) rank at most $r$ in a complex $(d+1)mn$ dimensional space is studied. For $r = 1, \dots , \min \{m, n\}-1$, we show that ${\cal P}^{m\times n}_{r,d}$ is the union of the closures of the $rd+1$ sets of matrix polynomials with rank $r$, degree exactly $d$, and explicitly described complete eigenstructures. In addition, for the full-rank rectangular polynomials, i.e. $r= \min \{m, n\}$ and $m \neq n$, we show that ${\cal P}^{m\times n}_{r,d}$ coincides with the closure of a single set of the polynomials with rank $r$, degree exactly $d$, and the described complete eigenstructure. These complete eigenstructures correspond to generic $m \times n$ matrix polynomials of grade $d$ and rank at most~$r$.

math.NA↗

arXiv2016

Miniversal deformations of pairs of skew-symmetric matrices under congruence

Miniversal deformations for pairs of skew-symmetric matrices under congruence are constructed. To be precise, for each such a pair $(A,B)$ we provide a normal form with a minimal number of independent parameters to which all pairs of skew-symmetric matrices $(\widetilde{A},\widetilde{B})$, close to $(A,B)$ can be reduced by congruence transformation which smoothly depends on the entries of the matrices in the pair $(\widetilde{A},\widetilde{B})$. An upper bound on the distance from such a miniversal deformation to $(A,B)$ is derived too. We also present an example of using miniversal deformations for analyzing changes in the canonical structure information (i.e. eigenvalues and minimal indices) of skew-symmetric matrix pairs under perturbations.

math.RT↗

arXiv2014

Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence

We construct the Hasse diagrams $G_2$ and $G_3$ for the closure ordering on the sets of congruence classes of $2\times 2$ and $3\times 3$ complex matrices. In other words, we construct two directed graphs whose vertices are $2\times 2$ or, respectively, $3\times 3$ canonical matrices under congruence and there is a directed path from $A$ to $B$ if and only if $A$ can be transformed by an arbitrarily small perturbation to a matrix that is congruent to $B$. A bundle of matrices under congruence is defined as a set of square matrices $A$ for which the pencils $A+λA^T$ belong to the same bundle under strict equivalence. In support of this definition, we show that all matrices in a congruence bundle of $2\times 2$ or $3\times 3$ matrices have the same properties with respect to perturbations. We construct the Hasse diagrams $G_2^{\rm B}$ and $G_3^{\rm B}$ for the closure ordering on the sets of congruence bundles of $2\times 2$ and, respectively, $3\times 3$ matrices. We find the isometry groups of $2\times 2$ and $3\times 3$ congruence canonical matrices.

math.RT↗