The Condition Number of Equivalence Transformations That Block Diagonalize Matrix Pencils

How ill-conditioned must a matrix S be if its columns are constrained to span certain subspaces? We answer this question in order to find nearly best conditioned matrices SR and SL that block diagonalize a given matrix pencil T = A + λ B, i.e. SL- 1 TSR = Θ is block diagonal. We show that the best conditioned SR has a condition number approximately equal to the cosecant of the smallest angle between right subspaces belonging to different diagonal blocks of Θ. Thus, the more nearly the right subspaces overlap the more ill-conditioned SR must be. The same is true of SL and the left subspaces. For the standard eigenproblem (T = A - λ I), SL = SR and the cosecant of the angle between subspaces turns out equal to an earlier estimate of the smallest condition number, namely the norm of the projection matrix associated with one of the subspaces. We apply this result to bound the error in an algorithm to compute analytic functions of matrices, for instance exp $(T)$.