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 this set. The new approach is based upon and meant to give insight into the staircase algorithm for the computation of the Jordan Canonical Form 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 pencil A -- lambdaB.
The Dimension of Matrices (Matrix Pencils) with Given Jordan (Kronecker) Canonical Forms
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