Let M n (F) denote the space of matrices over the eld F. Given A2 M n (F) deene jAj (A A) 1=2 and U(A) AjAj ?1 assuming A is nonsingular. Let 1 (A) 2 (A) n (A) 0 denote the ordered singular values of A. We obtain majorization results relating the singular values of U(A + A) ? U(A) and those of A and A. In particular we show that if A; A2 M n (R) and 1 ((A) < n (A) then for any unitarily invaria...

In this short paper, we establish a reverse of the derived inequalities for sector matrices by Tan and Xie, with Kantorovich constant. Then, as application our main theorem, some determinant unitarily invariant norm are presented.

Let A = UP be a polar decomposition of an n×n complex matrix A. Then for every unitarily invariant norm ||| · |||, it is shown that ||| |UP − PU |||| ≤ |||A∗A−AA∗||| ≤ ‖UP + PU‖ |||UP − PU |||, where ‖·‖ denotes the operator norm. This is a quantitative version of the wellknown result that A is normal if and only if UP = PU . Related inequalities involving self-commutators are also obtained.

We study complex-valued symmetric matrices. A simple expression for the spectral norm of such matrices is obtained, by utilizing a unitarily congruent invariant form. Consequently, we provide a sharp criterion for identifying those symmetric matrices whose spectral norm does not exceed one: such strongly stable matrices are usually sought in connection with convergent difference approximations ...

Let A,B be nonzero positive semidefinite matrices. We prove that ‖AB‖ ‖A‖ ‖B‖ ≤ ‖A + B‖ ‖A‖+ ‖B‖ , ‖A ◦B‖ ‖A‖ ‖B‖ ≤ ‖A + B‖ ‖A‖+ ‖B‖ for any unitarily invariant norm with ‖diag(1, 0, . . . , 0)‖ ≥ 1. Some related inequalities are derived. AMS classification: 15A60, 15A45

Let f(t) be a nonnegative concave function on 0 ≤ t < ∞ with f(0) = 0, and letX,Y be n×nmatrices. Then it is known that ‖f(|X+Y |)‖1 ≤ ‖f(|X|)‖1+‖f(|Y |)‖1, where ‖ · ‖1 is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.

We obtain a representation of unitarily invariant norm in terms of Ky Fan norms [1, p.35]. Indeed we obtain a more general result in the context of Eaton triple with reduced triple. Examples are given. 2000 Mathematics Subject Classification: 15A60, 65F35.

Let A − λB and C − λD be two normal matrix pencils or Hermitian matrix pencils or definite matrix pencils, and let AXD −BXC = S. Our main results in this part are estimates of a unitarily invariant norm of the solution X under some conditions on the spectra of two pencils. §

Let m,n, p be positive integers such that m ≥ n + p. Suppose (A,B) ∈ Cm×n ×Cm×p, and let P(A,B) = {(E,F ) ∈ Cm×n ×Cm×p : there is X ∈ Cn×p such that (A− E)X = B − F}. The total least square problem concerns the determination of the existence of (E,F ) in P(A,B) having the smallest Frobenius norm. In this paper, we characterize elements of the set P(A,B) and derive a formula for ρ(A,B) = inf {‖[...