Browsing by Author "Chang, Chi-Tung"

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Equality of higher-rank numerical ranges of matrices
(2014-05-04)
Chang, Chi-Tung
;
Gau, Hwa-Long
;
Wang, Kuo-Zhong
;
應用數學系
;
Department of Applied Mathematics
Let denote the rank- numerical range of an -by- complex matrix . We give a characterization for , where , via the compressions and the principal submatrices of . As an application, the matrix satisfying , where is the classical numerical range of and , is under consideration. We show that if for some , then is unitarily similar to , where is a 2-by-2 matrix, is a -by- matrix and .
Maximizing numerical radii of weighted shifts under weight permutations
(2012-10-15)
Chang, Chi-Tung
;
Wang, Kuo-Zhong
;
應用數學系
;
Department of Applied Mathematics
Let omega(i) is an element of C (1 <= i <= n) and I is an element of S-n, the symmetric group of all permutations of 1, 2,...,n. Suppose A(I) is the weighted cyclic matrix [GRAPHICS] and omega(A(I)) denotes its numerical radius. We characterize those zeta is an element of S-n which satisfy omega(A(zeta)) = max(vertical bar is an element of Sn) omega(A(I)). The characterizations for unilateral and bilateral weighted (backward) shifts are also obtained. Crown Copyright (C) 2012 Published by Elsevier Inc. All rights reserved.
Numerical ranges and Gersgorin discs
(2013-02-01)
Chang, Chi-Tung
;
Gau, Hwa-Long
;
Wang, Kuo-Zhong
;
Wu, Pei Yuan
;
應用數學系
For a complex matrix A = vertical bar a(ij)vertical bar(n)(i,j=1), let W(A) be its numerical range, and let G(A) be the convex hull of U-i=1(n) {z is an element of C : vertical bar z - a(ij)vertical bar <= (Sigma(i not equal j)(vertical bar a(ij)vertical bar)/2} and G'(A) = n{G(U*AU) : U n-by-n unitary). It is known that W(A) is always contained in G(A) and hence in G'(A). In this paper, we consider conditions for W (A) to be equal to G(A) or G'(A). We show that if W(A) = G'(A). then the boundary of W(A) consists only of circular arcs and line segments. If, moreover, A is unitarily irreducible, then W (A) is a circular disc. (Almost) complete characterizations of 2-by-2 and 3-by-3 matrices A for which W(A) = G'(A) are obtained. We also give criteria for the equality of W(A) and G(A). In particular, such A's among the permutationally irreducible ones must have even sizes. We also characterize those A's with size 2 or 4 which satisfy W(A) = G(A). Crown Copyright (C) 2012 Published by Elsevier Inc. All rights reserved.