Reordering Diagonal Blocks In Real Schur Form

We present a direct algorithm for computing an orthogonal similarity transformation which interchanges neighboring diagonal blocks in a matrix in real Schur form. The algorithm does not require the solution of the associated Sylvester equation. Numerical tests suggest the backward stability of the scheme. 1 Introduction The problem of reordering eigenvalues of a matrix in real Schur form arises in the computation of the invariant subspaces corresponding to a group of eigenvalues of the matrix. A basic step in such reordering is to swap two neighboring 1 \Theta 1 or 2 \Theta 2 diagonal blocks by an orthogonal transformation. Swapping two 1 \Theta 1 blocks or swapping 1 \Theta 1 and 2 \Theta 2 blocks are well understood [3]. Swapping two 2 \Theta 2 blocks poses some numerical difficulties. Recently, Bai and Demmel [1] have proposed an algorithm for swapping two 2 \Theta 2 blocks which is for all practical purposes backward stable. The algorithm requires the solution of a Sylvester equat..