Research Seminar in Mathematics - Reducing collections of matrices

Speaker

Andrii Dmytryshyn, Umeå universitet.

Abstract

We present Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and $*$-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 $2 \times 2$ block matrix representations of the equations are block diagonalizable by linked equivalence transformations. 

We also describe the class of all possible sets of complex matrices that can be reduced to an upper-triangular form by associated unitary transformations, and the class of all possible sets of real matrices that can be reduced to a quasi-upper-triangular form by associated orthogonal transformations. Here one may think of Schur forms for a single matrix, a matrix pencil, and matrices associated with the periodic eigenvalue problem which all are frequently used and studied representatives of this class. Schur forms are the key ingredient for solving systems of Sylvester matrix equations by generalizations of the Bartels-Stewart algorithm.