Componentwise Perturbation Analysis Of The Schur Decomposition Of A Matrix

This paper presents a unified scheme for perturbation analysis of the Schur decomposition A = UTUH of an nth order matrix A which allows one to obtain new local (asymptotic) componentwise perturbation bounds for the corresponding unitary transformation matrix U and upper triangular matrix T. This scheme involves n(n -1)/2 basic perturbation parameters which determine all componentwise bounds for the elements of U, the eigenvalues, the invariant subspaces, and the superdiagonal elements of T. New sensitivity estimates and condition numbers of eigenvalues, invariant subspaces, and superdiagonal elements are derived which produce theoretically the same results but are computationally alternative to the well-known local perturbation bounds. These estimates are based on computing the inverse of a block lower triangular matrix of order n(n -1)/2, which is obtained from the Schur form, and do not involve eigenvectors. Since the computation of the inverse may be done efficiently by parallel algorithms, the implementation of new estimates can be advantageous in comparison with the usage of classical estimates.