To Örebro University

We investigate the changes of the canonical structure information under small perturbations for a system pencil associated with a (generalized) linear time-invariant state-space system. The equivalence class of the pencil is taken with respect to feedback-injection equivalence transformations. The results allow us to track possible changes of important linear system characteristics under small perturbations.

We investigate the changes under small perturbations of the canonical structure information for a system pencil (A B C D) − s (E 0 0 0), det(E) ≠ 0, associated with a (generalized) linear time-invariant state-space system. The equivalence class of the pencil is taken with respect to feedback-injection equivalence transformation. The results allow to track possible changes under small perturbations of important linear system characteristics.

We construct the Hasse diagrams G₂ and G₃ for the closure ordering on the sets of congruence classes of 2 × 2 and 3 × 3 complex matrices. In other words, we construct two directed graphs whose vertices are 2 × 2 or, respectively, 3 × 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 × 2 or 3 × 3 matrices have the same properties with respect to perturbations. We construct the Hasse diagrams G₂ ^B and G₃ ^B for the closure ordering on the sets of congruence bundles of 2 × 2 and, respectively, 3 × 3 matrices. We find the isometry groups of 2 × 2 and 3 × 3 congruence canonical matrices.

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

We study how small perturbations of matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy graphs (stratifications) of orbits and bundles of matrix polynomial Fiedler linearizations. We show that the stratifica-tion graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler lineariza-tions have the same geometry (topology). The results are illustrated by examples using the software tool StratiGraph.

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 A - lambda B can be approximated by pencils strictly equivalent to a skew-symmetric matrix pencil C - lambda D if and only if A - lambda B can be approximated by pencils congruent to C - lambda D.

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. This theory relies on our main theorem stating that a skew-symmetric matrix pencil A-λB can be approximated by pencils strictly equivalent to a skew-symmetric matrix pencil C-λD if and only if A-λB can be approximated by pencils congruent to C-λD.

The set of all solutions to the homogeneous system of matrix equations (X-T A + AX, X-T B + BX) = (0, 0), where (A, B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A, B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. Kagstrom, and V. V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375-3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.

Matlab functions to work with the canonical structures for congru-ence and *congruence of matrices, and for congruence of symmetricand skew-symmetric matrix pencils are presented. A user can providethe canonical structure objects or create (random) matrix examplesetups with a desired canonical information, and compute the codi-mensions of the corresponding orbits: if the structural information(the canonical form) of a matrix or a matrix pencil is known it isused for the codimension computations, otherwise they are computednumerically. Some auxiliary functions are provided too. All thesefunctions extend the Matrix Canonical Structure Toolbox.

The homogeneous system of matrix equations (X(T)A + AX, (XB)-B-T + BX) = (0, 0), where (A, B) is a pair of skew-symmetric matrices of the same size is considered: we establish the general solution and calculate the codimension of the orbit of (A, B) under congruence. These results will be useful in the development of the stratification theory for orbits of skew-symmetric matrix pencils.