Probabilistic analysis of an estimator for the Frobenius norm of a matrix product

Abstract

Estimating the Frobenius norm of a matrix product C=XY without computing C explicitly is required in applications such as the one-sided block Jacobi method. In this paper, we analyze Bečka et al.'s estimator for this problem within a probabilistic framework. Specifically, we consider the set of matrices with the Frobenius norm $\|C\|_F^2$ and introduce some natural probability measure into it. Then, we show that if we choose a matrix randomly from this set and apply the estimator, the expected value of the square of this estimator is exactly $\|C\|_F^2$.