Structured Condition Number and Backward Error for Eigenvalue Problems

In this paper, we investigate condition number and backward error for eigenvalue problems. Results on unstructured condition number for a simple eigenvalue are recalled and then a definition of a structured condition number is given for linear structures that are Toeplitz, circulant, Hankel, symmetric, Hermitian and skew-Hermitian. For these structures (except for circulant), we show that the unstructured condition number equals the structured condition number. We generalize these results to eigenvalues of matrix polynomials. We also study structured backward error for matrix polynomials.