In 1937, von Neumann [31] gave a famous characterization of unitarily invariant matrix norms (that is, norms f on Cp×q satisfying f(uxv) = f(x) for all unitary matrices u and v and matrices x in Cp×q). His result states that such norms are those functions of the form g ◦ , where the map x ∈ Cp×q 7→ (x) ∈ R has components the singular values 1(x) ≥ 2(x) ≥ · · · ≥ p(x) of x (assuming p ≤ q) and g...

Let A be an m n (m n) complex matrix. It is known that there is a unique polar decomposition A = QH, where Q Q = I, the n n identity matrix, and H is positive de nite, provided A has full column rank. This note addresses the following question: how much may Q change if A is perturbed? For the square case m = n our bound, which is valid for any unitarily invariant norm, is sharper and simpler th...

Low rank matrix approximations have many applications in different domains. In system theory it has been used in model reduction schemes, in system identification with outputerror models and in static errors-in-variables problems, for instance. The approximations are mostly performed using the singular value decomposition. This is optimal for all unitarily invariant matrix norms, such as the Fr...

In this paper, we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities, it is shown that if A, B, X are [Formula: see text] matrices, then [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are non-negative continuous functions such that [Formula: see text] and [Formula: see...