TABLE OF CONTENTS While it's not always possible to diagonalize a matrix A ? C m m with a similarity transformation, Schur's theorem (p. 508) guarantees that every A ? C m m is unitarily similar to an upper-triangular matrix-say U* AU = T. But other than the fact that the diagonal entries of T are the eigenvalues of A, there is no pattern to the nonzero part of T . So to what extent can this be remedied by giving up the unitary nature of U? In other words, is there a nonunitary P for which P ?1 AP has a simpler and more predictable pattern than that of T? We have already made the first step in answering this question. The core-nilpotent decomposition (p. 397) says that for every singular matrix A of index k and rank r, there is a nonsingular matrix Q such that
Consequently, any further simplification by means of similarity transformations can revolve around C and L . Let's begin by examining the degree to which nilpotent matrices can be reduced by similarity transformations.
In what follows, let L n n be a nilpotent matrix of index k so that L k = 0 but L k ?1 ? 0 . The first question is, "Can L be diagonalized by a similarity transformation?" To answer this, notice...
