Graph-theoretically determined Jordan-block-size structure of regular matrix pencils

Abstract

The authors investigate the sizes of Jordan blocks of regular matrix pencils by means of a one-to-one correspondence between a matrix pencil ( λE + μA) and a weighted digraph G( E, A). Based on the relationship between determinantal divisors of a pencil and spanning-cycle families of the associated digraph G( E, A), the Jordan-block-size structure is determined graph-theoretically. For classes of structurally equivalent matrix pencils defined by a pair of structure matrices [ E, A], the generic Jordan block sizes corresponding to the characteristic roots at zero and at infinity can be obtained from the unweighted digraph G([ E], [ A]). Eigenvalues of matrices are discussed as special cases. A nontrivial mechanical example illustrates the procedure.