A matrix minimization problem involving ranks
Here one considers the following minimization problem: given a finite, not necessarily square real matrix \(Q^ 0\), and \(k<rank Q^ 0\), to find a real matrix Q of the same size as \(Q^ 0\), such that the norm of \(Q- Q^ 0\) is minimized under the condition that rank \(Q\leq k\). One approaches the problem by using generalized gradients and normal cones introduced by \textit{F. H. Clarke} [Optimization and nonsmooth analysis (1983; Zbl 0582.49001)] and investigated by him, Rockafellar and others. This approach is quite different from the usual approaches, one of which is by applying results about singular values of \(Q^ 0\) of \textit{G. W. Stewart} [Introduction to matrix computations (1973; Zbl 0302.65021)]. Instead of the \(\ell_ 2\)-induced norm convenient for the approach using singular values one takes here the Frobenius norm. Special attention is devoted to matrices \(Q^ 0\) and Q which are finite Hankel matrices of special nature and called here reversely circulant. Some of the results are also valid for real matrices with a countably infinite number of rows and columns. It may happen that the rank of the minimizing Q is less than k, hence requiring that rank Q\(=k\) would lead to another problem.
scientific article; zbMATH DE number 4006450
scientific article