THE DIMENSION OF MATRICES (MATRIX PENCILS) WITH GIVEN JORDAN (KRONECKER) CANONICAL-FORMS

The set of n by n matrices with a given Jordan canonical form defines a subset of matrices in complex n(2) dimensional space. We analyze one classical approach and one new approach to count the dimension of thi s set. The new approach is based upon and meant to give insight into t he staircase algorithm for the computation of the Jordan canonical for m as well as the occasional failures of this algorithm. We extend both techniques to count the dimension of the more complicated set defined by the Kronecker canonical form of an arbitrary rectangular matrix pe ncil A - lambda B.