In this appendix, we prove Theorem 4 and Lemmas 7 – 12 in order. A.1. Proof of Theorem 4. We first need a lemma for perturbation bound of square root matrices. Lemma 16. Let A, B be positive semi-definite matrices, and then for any unitarily invariant norm ï¿¿·ï¿¿, ï¿¿A 1/2 − B 1/2 ï¿¿ ≤ 1 σ min (A 1/2) + σ min (B 1/2) ï¿¿A − Bï¿¿. Proof. The proof essentially follows the idea of [27]. Let D = ...

Given matrices of the same size, A = a ij ] and B = b ij ], we deene their Hadamard Product to be A B = a ij b ij ]. We show that if x i > 0 and q p 0 then the n n matrices q j # are positive deenite and relate these facts to some matrix valued arithmetic-geometric-harmonic mean inequalities-some of which involve Hadamard products and others unitarily invariant norms. It is known that if A is p...

Low-rank matrix recovery is an active topic drawing the attention of many researchers. It addresses the problem of approximating the observed data matrix by an unknown low-rank matrix. Suppose that A is a low-rank matrix approximation of D, where D and A are [Formula: see text] matrices. Based on a useful decomposition of [Formula: see text], for the unitarily invariant norm [Formula: see text]...

‎this paper presents two main results that the singular values of the hadamard product of normal matrices $a_i$ are weakly log-majorized by the singular values of the hadamard product of $|a_{i}|$ and the singular values of the sum of normal matrices $a_i$ are weakly log-majorized by the singular values of the sum of $|a_{i}|$‎. ‎some applications to these inequalities are also given‎. ‎in addi...

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. ...

Let ‖| · ‖| be any give unitarily invariant norm. We obtain some exponential relations in the context of semisimple Lie group. On one hand they extend the inequalities (1) ‖|e‖| ≤ ‖|eReA‖| for all A ∈ Cn×n, where ReA denotes the Hermitian part of A, and (2) ‖|e‖| ≤ ‖|ee‖|, where A and B are n×n Hermitian matrices. On the other hand, the inequalities of Weyl, Ky Fan, Golden-Thompson, Lenard-Thom...

Let Gm,n be the Grassmann space of m-dimensional subspaces of F. Denote by θ1(X ,Y), . . . , θm(X ,Y) the canonical angles between subspaces X ,Y ∈ Gm,n. It is shown that Φ(θ1(X ,Y), . . . , θm(X ,Y)) defines a unitarily invariant metric on Gm,n for every symmetric gauge function Φ. This provides a wide class of new metrics on Gm,n. Some related results on perturbation and approximation of subs...