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...

The unitarily invariant norms of matrices, or operators, are essentially the symmetric norms of their singular values. A subclass of these norms depending upon only a few largest of the singular values is considered, and the polars of these norms are characterized. The result is then used to obtain generalizations of some well-known inequalities. The implications for operators on infinite-dimen...

The singular value decomposition (SVD) is not only a classical theory in matrix computation and analysis, but also is a powerful tool in machine learning and modern data analysis. In this tutorial we first study the basic notion of SVD and then show the central role of SVD in matrices. Using majorization theory, we consider variational principles of singular values and eigenvalues. Built on SVD...