Tomaž Košir

ON SEMITRANSITIVE JORDAN ALGEBRAS OF MATRICES

A set

The Cayley-Hamilton theorem and inverse problems for multiparameter systems

\mathcal{S}

On stability of invariant subspaces of commuting matrices

of linear operators on a vector space is said to be semitransitive if, given nonzero vectors x, y, there exists

Geometric aspects of multiparameter spectral theory

A\in \mathcal{S}

Common fixed points and common eigenvectors for sets of matrices

such that either Ax = y or Ay = x. In this paper we consider semitransitive Jordan algebras of operators on a finite-dimensional vector space over an algebraically closed field of characteristic not two. Two of our main results are: (1) Every irreducible semitransitive Jordan algebra is actually transitive. (2) Every semitransitive Jordan algebra contains, up to simultaneous similarity, the upper triangular Toeplitz algebra, i.e. the unital (associative) algebra generated by a nilpotent operator of maximal index.

Common Jordan chains of matrices

Abstract We review some of the current research in multiparameter spectral theory. We prove a version of the Cayley–Hamilton theorem for multiparameter systems and list a few inverse problems for such systems. Some consequences of results on determinantal representations proved by Dixon, Dickson and Vinnikov for the inverse problems are discussed.

Determinantal representations of smooth cubic surfaces

We study the stability of (joint) invariant subspaces of a finite set of commuting matrices. We generalize some of the results of Gohberg, Lancaster, and Rodman for the single matrix case. For sets of two or more commuting matrices we exhibit some phenomena different from the single matrix case. We show that each root subspace is a stable invariant subspace, that each invariant subspace of a root subspace of a nonderogatory eigenvalue is stable, and that, even in the derogatory case, the eigenspace is stable if it is one-dimensional. We prove that a pair of commuting matrices has only finitely many stable invariant subspaces. At the end, we discuss the stability of invariant subspaces of an algebraic multiparameter eigenvalue problem.

On pairs of commuting nilpotent matrices

The paper contains a geometric description of the dimension of the total root subspace of a regular multiparameter system in terms of the intersection multiplicities of its determinantal hypersurfaces. The new definition of regularity used is proved to restrict to the familiar definition in the linear case. A decomposability problem is also considered. It is shown that the joint root subspace of a multiparameter system may be decomposable even when the root subspace of each member is indecomposable.

A Groebner basis for the 2×2 determinantal ideal mod t2☆

The following questions are studied: Under what conditions does the existence of a (nonzero) fixed point for every member of a semigroup of matrices imply a common fixed point for the entire semigroup? What is the smallest number k such that the existence of a common fixed point for every k members of a semigroup implies the same for the semigroup? If every member has a fixed space of dimension at least k: What is the best that can be said about the common fixed space? We also consider analogs of these questions with general eigenspaces replacing fixed spaces.

Irreducible semigroups of matrices with eigenvalue one

A necessary and sufficient condition for any set of matrices to have an eigenvector in common is given; and the connection between common divisors of matrix polynomials and common Jordan chains of their first companion matrices is studied.