On the Relation of Schatten Norms and the Thompson Metric

The Thompson metric provides key geometric insights in the study or non-linear matrix equations and in many optimization problems. However, knowing that an approximate solution is within d_T units of the actual solution in the Thompson metric provides little insight into how good the approximation is as a matrix or vector approximation. That is, bounding the Thompson metric between an approximate and accurate solution to a problem does not provide obvious bounds either for the spectral or the Frobenius norm, both Schatten norms, of the difference between the approximation and accurate solution. This paper reports an upper bound on the Schatten norm of X - Y related to both the Thompson metric between X and Y and the maximum of their Schatten norms. This paper reports a similar but slightly tighter bound for the Frobenius norm of X - Y.