Voiculescu’s asymptotic freeness result for random matrices is improved to the sense of almost everywhere convergence. The asymptotic freeness almost everywhere is first shown for standard unitary matrices based on the computation of multiple moments of their entries, and then it is shown for rather general unitarily invariant selfadjoint random matrices (in particular, standard selfadjoint Gau...

We give a characterization of all the unitarily invariant norms on finite von Neumann algebra acting separable Hilbert space. The is analogous to Neumann’s for $$n\times n$$ complex matrices and in Fang et al. (J Funct Anal 255(1):142–183, 2008) $$II_{1}$$ factors.

We prove an inequality for unitarily invariant norms that interpolates between the Arithmetic-Geometric Mean inequality and the Cauchy-Schwarz inequality.

Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ide...

Here, we will consider one approach for extending the ideas underlying the least-squares algorithm we discussed in class to non-tall matrices. Let A ∈ Rn×d matrix, where both n and d are large, and where rank(A) = k exactly, and let B ∈ Rn×t. Consider the problem minX∈Rn×t‖AX −B‖, where ‖ · ‖ is a unitarily-invariant matrix norm. The solution to this problem is Xopt = AB, and here we consider a...