Browsing by Author "Gau, HL"

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Companion matrices: reducibility, numerical ranges and similarity to contractions
(2004-05-15)
Gau, HL
;
Pei, YW
;
應用數學系
;
Department of Applied Mathematics
In this paper, we study some unitary-equivalence properties of the companion matrices. We obtain a criterion for a companion matrix to be reducible and show that the numerical range of a companion matrix is circular disc centered at the origin if and only if the matrix equals the (nilpotent) Jordan block. However, the more general assertion that a companion matrix is determined by its numerical range turns out to be false. We also determine, for an n x n matrix A with eigenvalues in the open unit disc, the defect index of a contraction to which A is similar. (C) 2004 Elsevier Inc. All rights reserved.
Condition for the numerical range to contain an elliptic disc
(2003-05-01)
Gau, HL
;
Wu, PY
;
應用數學系
;
Department of Applied Mathematics
For an n-by-n matrix A and an elliptic disc E in the plane, we show that the sum of the number of common supporting lines and the number of common intersection points to E and the numerical range W (A) of A should be at least 2n + 1 in order to guarantee that E be contained in W(A). This generalizes previous results of Anderson and Thompson. As an application, our result is used to verify a special case of the Poncelet property conjecture. (C) 2003 Elsevier Science Inc. All rights reserved.
Finite Blaschke products of contractions
(2003-07-15)
Gau, HL
;
Wu, PY
;
應用數學系
;
Department of Applied Mathematics
Let A be a contraction on Hilbert space H and phi a finite Blaschke product. In this paper, we consider the problem when the norm of phi(A) is equal to 1. We show that (1) parallel tophi(A)parallel to = 1 if and only if parallel toA(k)parallel to = 1, where k is the number of zeros of phi counting multiplicity, and (2) if H is finite-dimensional and A has no eigenvalue of modulus 1, then the largest integer I for which parallel toA(l)parallel to = 1 is at least m/(n - m), where n = dim H and m = dim ker(I - A*A), and, moreover, l = n - 1 if and only if m = n - 1. (C) 2003 Elsevier Science Inc. All rights reserved.
Numerical range and Poncelet property
(2003-06-01)
Gau, HL
;
Wu, PY
;
應用數學系
;
Department of Applied Mathematics
In this survey article, we give an expository account of the recent developments on the Poncelet property for numerical ranges of the compressions of the shift S(phi). It can be considered as an updated and more advanced edition of the recent expository article published in the American Mathematical Monthly by the second author on this topic. The new information includes: (1) a simplified approach to the main results (generalizations of Poncelet, Brianchon-Ceva and Lucas-Siebeck theorems) in this area, (2) the recent discovery of Mirman reftiting a previous conjecture on the coincidence of Poncelet curves and boundaries of the numerical ranges of finite-dimensional S(phi), and (3) some partial generalizations by the present authors of the above-mentioned results from the unitary-dilation context to the normal-dilation one and also from the finite-dimensional S(phi) to the infinite-dimensional.
Numerical range circumscribed by two polygons
(2004-05-01)
Gau, HL
;
Wu, PY
;
應用數學系
;
Department of Applied Mathematics
We show that, for any 2n+2 distinct points a(1), a'(1), a(2), a'(2),...,a(n+1), a'(n+1) (in this order) on the unit circle, there is an n-by-n matrix A. unique LIP to Unitary equivalence, which has norm one and satisfies the conditions that it has all its eigenvalues in the open unit disc, I-n-A*A has rank one and its numerical range is circumscribed by the two (n+1)-gons a(1)a(2)...a(n+1) and a'(1)a'(2)...a'(n+1). This generalizes the classical result of the existence of a conical Curve Circumscribed by two triangles which are already inscribed on another conical curve. (C) 2004 Elsevier Inc. All rights reserved.
Numerical range of a normal compression
(2004-05-01)
Gau, HL
;
Wu, PY
;
應用數學系
;
Department of Applied Mathematics
Let N be an n-by-n diagonal matrix whose distinct eigenvalues form corners of their convex hull, let x be a vector in with nonzero components, and let A be the compression of N to the orthogonal complement of x . In this article, we study the properties of the eigenvalues and the numerical range of A and show that in many ways they are analogous to the ones for unitary N . The approach via the diagonal form of N yields a much simpler proof for some of the main results in this area.
Numerical range of a normal compression II
(2004-10-01)
Gau, HL
;
Wu, PY
;
應用數學系
;
Department of Applied Mathematics
As in the predecessor [Numerical range of a normal compression, Linear and Multilinear Algebra, in press] of this paper, we consider properties of matrices of the form V*NV, where N = diag(a(1),..., a(n+1)) is a diagonal matrix with distinct eigenvalues a(j)s such that each of them is a corner of the convex hull they generate, and V is an (n + 1)-by-n matrix with V*V = I-n such that any nonzero vector orthogonal to the range space of V has all its components nonzero. We obtain that such a matrix A is determined by its eigenvalues up to unitary equivalence, is irreducible and cyclic, and the boundary of its numerical range is a differentiable curve which contains no line segment. We also consider the condition for the existence of another matrix of the above type which dilates to A such that their numerical ranges share some common points with the boundary of the (n + 1)-gon a(1) (...) a(n+1). (C) 2004 Elsevier Inc. All rights reserved.