This is the first paper of a two-long series in which we study linear generalized inverses that minimize matrix norms. Such generalized inverses are famously represented by the Moore-Penrose pseudoinverse (MPP) which happens to minimize the Frobenius norm. Freeing up the degrees of freedom associated with Frobenius optimality enables us to promote other interesting properties. In this Part I, w...

In this paper, we present some refinements of reverse Young?s inequalities. Among other results, a refinement operator Young inequalities says A?vB + 2?(A?B ? A?B) m??M / A?vB, where 0 < mI A, B MI, = min{v, 1 v} and v [0, 1], extending key result in [J. Math. Anal. Appl. 465 (2018) 267-280] [Linear Multilinear Algebra 67 (2019) 1567-1578]. Furthermore, give due to [Math. Slovaca 70 (2020), ...

We design new sketching algorithms for unitarily invariant matrix norms, including the Schatten p-norms ‖·‖Sp , and obtain, as a by-product, streaming algorithms that approximate the norm of a matrix A presented as a turnstile data stream. The primary advantage of our streaming algorithms is that they are simpler and faster than previous algorithms, while requiring the same or less storage. Our...

Low-rank inducing unitarily invariant norms have been introduced to convexify problems with low-rank/sparsity constraint. They are the convex envelope of a unitary norm and indicator function an upper bounding rank The most well-known member this family is so-called nuclear norm. To solve optimization involving such proximal splitting methods, efficient ways evaluating mapping low-rank needed. ...

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...

Suppose U is an upper-triangular matrix, and D a nonsingular diagonal matrix whose diagonal entries appear in nondescending order of magnitude down the diagonal. It is proved that kD UDk kUk for any matrix norm that is reduced by a pinching. In addition to known examples { weakly unitarily invariant norms { we show that any matrix norm de ned by kAk def = max x 6=0; y 6=0 Re (x Ay) (x) (y) ; wh...

We study matrix inequalities involving partial traces for positive semidefinite block matrices. First of all, we present a new method to prove celebrated result Choi [Linear Algebra Appl. 516 (2017)]. Our also allows us generalization another Multilinear 66 (2018)]. Furthermore, shall give an improvement on recent Li, Liu and Huang [Operators Matrices 15 (2021)]. In addition, include with some ...