Andrii Dmytryshyn

Miniversal deformations of matrices of bilinear forms

Arnold [V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26 (2) (1971) 29–43] constructed miniversal deformations of square complex matrices under similarity; that is, a simp ...

COUPLED SYLVESTER-TYPE MATRIX EQUATIONS AND BLOCK DIAGONALIZATION ∗

We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and

Miniversal deformations of matrices under *congruence and reducing transformations

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Orbit closure hierarchies of skew-symmetric matrix pencils

-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of

SYMMETRIC MATRIX PENCILS: CODIMENSION COUNTS AND THE SOLUTION OF A PAIR OF MATRIX EQUATIONS

2 \times 2

Canonical Structure Transitions of System Pencils

block matrix representations of the equations are block diagonalizable by (linked) equivalence transformations. Various applications leading to several particular cases have already been investigated in the literature, some recently and some long ago. Solvability of these cases follow immediately from our general consistency theory. We also show how to apply our main result to systems of Stein-type matrix equations.

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

Arnold (1971) [1] constructed a miniversal deformation of a square complex matrix under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We give miniversal deformations of matrices of sesquilinear forms; that is, of square complex matrices under *congruence, and construct an analytic reducing transformation to a miniversal deformation. Analogous results for matrices under congruence were obtained by Dmytryshyn, Futorny, and Sergeichuk (2012) [11].

Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations

We study how small perturbations of a skew-symmetric matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of skew-symmetric matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a skew-symmetric matrix pencil contains the congruence orbit (or bundle) of another skew-symmetric matrix pencil. The developed theory relies on our main theorem stating that a skew-symmetric matrix pencil

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

A-\lambda B

Structure preserving stratification of skew-symmetric matrix polynomials

can be approximated by pencils strictly equivalent to a skew-symmetric matrix pencil