Numerical Analyst's Jordan Canonical Form.

摘要

What does it mean to compute an eigendecomposition of an uncertain matrix. Because of measurement errors and roundoff errors, one must typically compute the eigenvalues and eigenvectors not of a single matrix but rather of a ball of matrices whose radius depends on the uncertainty in the data. We approach this problem by asking how to partition the eigenvalues of the matrices in the ball into nonoverlapping groups which cannot themselves be further partitioned. More specifically, we define the dissociation of two subsets sigma sub 1 and sigma sub 2 = sigma x sigma sub 1 of the sets of eigenvalues sigma of a matrix T as the smallest perturbation of T that will make some eigenvalue from sigma sub 1 and some eigenvalue from sigma sub 2 move together and become indistinguishable. The results of this thesis are of two kinds. First, we compute upper and lower bounds on the dissociation which improve bounds in the literature. Both upper and lower bounds are achievable or nearly so. The upper and lower bounds are often close together but occasionally far apart. Our second set of results quantifies this last statement by assuming a probability density on the set of matrices and computing the likelihood that the bounds are far apart. This approach leads to numerous other probabilistic results, such as the distribution of the condition number of a random matrix, and the distribution of the distance from a random matrix to one with a given Jordan form.