Linear relations and the Kronecker canonical form
We show that the Kronecker canonical form (which is a canonical decomposition for pairs of matrices) is the representation of a linear relation in a finite dimensional space. This provides a new geometric view upon the Kronecker canonical form. Each of the four entries of the Kronecker canonical form has a natural meaning for the linear relation which it represents. These four entries represent the Jordan chains at finite eigenvalues, the Jordan chains at infinity, the so-called singular chains and the multi-shift part. Or, to state it more concise: For linear relations the Kronecker canonical form is the analogue of the Jordan canonical form for matrices.
MSC 2010: Primary: 15A21 Canonical forms, reductions, classification 47A06 Linear relations (multivalued linear operators) Secondary: 15A22 Matrix pencils 47A15 Invariant subspaces