The conditionality and expected error of two methods of computing the pseudo-eigenvalues of a complex matrix

A relation is established between the conditionalities for two methods of calculating the pseudo-eigenvalues of a complex matrix A: a method in which the matrix AA (or AA) is formed explicitly, and a method in which a pseudo-QR-algorithm is applied to A itself. In certain cases (such as that of complex symmetric matrices) the absolute errors of small pseudo-eigenvalues calculated by the first method are much larger.