Staircase failures explained by orthogonal versal forms

Treating matrices as points in n²-dimensional space, we apply geometry to study and explain algorithms for the numerical determination of the Jordan structure of a matrix. Traditional notions such as sensitivity of subspaces are replaced with angles between tangent spaces of manifolds in n²-dimensional space. We show that the subspace sensitivity is associated with a small angle between complementary subspaces of a tangent space on a manifold in n²-dimensional space. We further show that staircase algorithm failure is related to a small angle between what we call staircase invariant space and this tangent space. The matrix notions in n²-dimensional space are generalized to pencils in 2mn-dimensional space. We apply our theory to special examples studied by Boley, Demmel, and Kågström.