Relative perturbation bounds for the unitary polar factor
For each \(m\times n\) complex (or real) matrix \(B\) \((m\geq n)\) with full column rank we have a unique polar decomposition \(B=QH\), where \(Q\) has orthonormal columns and \(H\) is positive definite. The following problem is discussed: How much may \(Q\) change if \(B\) is perturbed to \(\widetilde B= D^*_1BD_2\), where \(D^*_1\) and \(D_2\) are nonsingular and ``close'' to the identity matrix of the same size. It is shown that the change in \(Q\) is bounded only by the distances of \(D^*_1\) and \(D_2\) to the corresponding identity matrices, but does not depend on the singular values of \(B\).
scientific article; zbMATH DE number 1000364