A Jordan-like decomposition for linear relations in finite-dimensional spaces

TL;DR

This paper extends the classical Jordan canonical form to linear relations in finite-dimensional spaces, introducing a richer decomposition with four chain types and establishing its uniqueness.

Contribution

It develops a Jordan-like decomposition for linear relations, including new chain types and a unified algebraic framework with a uniqueness result.

Findings

Decomposition includes singular, Jordan, eigenvalue infinity, and multishift parts.

The structure is uniquely determined by Weyr characteristics.

The approach is purely algebraic and uniform across chain types.

Abstract

A square matrix $A$ has the usual Jordan canonical form that describes the structure of $A$ via eigenvalues and the corresponding Jordan blocks. If $A$ is a linear relation in a finite-dimensional linear space ${\mathfrak H}$ (i.e., $A$ is a linear subspace of ${\mathfrak H} \times {\mathfrak H}$ and can be considered as a multivalued linear operator), then there is a richer structure. In addition to the classical Jordan chains (interpreted in the Cartesian product ${\mathfrak H} \times {\mathfrak H}$), there occur three more classes of chains: chains starting at zero (the chains for the eigenvalue infinity), chains starting at zero and also ending at zero (the singular chains), and chains with linearly independent entries (the shift chains). These four types of chains give rise to a direct sum decomposition (a Jordan-like decomposition) of the linear relation $A$. In this decomposition there is a completely singular part that has the extended complex plane as eigenvalues; a usual Jordan part that corresponds to the finite proper eigenvalues; a Jordan part that corresponds to the eigenvalue $\infty$; and a multishift, i.e., a part that has no eigenvalues at all. Furthermore, the Jordan-like decomposition exhibits a certain uniqueness, closing a gap in earlier results. The presentation is purely algebraic, only the structure of linear spaces is used. Moreover, the presentation has a uniform character: each of the above types is constructed via an appropriately chosen sequence of quotient spaces. The dimensions of the spaces are the Weyr characteristics, which uniquely determine the Jordan-like decomposition of the linear relation.