| 摘要: | 研究期間：10108~10207;In this project, we study the existence and number of line segments on the boundaries of numerical ranges of various matrix classes. where the aj's, called the weights of A, are complex numbers. Let A[j] denote the (n — 1)-by-(n — 1) principal submatrix of A obtained by deleting its j th row and j th column. We want to show that dW (A) has a line segment if and only if the aj's are nonzero and W(A[k]) = W(A[/]) = W(A[m]) for some k, I and m, 1 < k <1 < m < n. In this case, W(A[j]) is the circular disc centered at the origin for all 1 < j < n, and there are exactly n line segments on dW (A). This will refine and extend the result of Tsai and Wu. Moreover, we conjecture that A and B are two n-by-n (n > 3) weighted shift matrices. Assume that the weights of A are nonzero and periodic with period k. If W(A) = W(B), then the weights of B are also nonzero and periodic with period k. Recall that, for any complex polynomial p(z) = zn + aizn-1 + ••• + an_iz + a,n, there is associated an n-by-n matrix called the companion matrix of p and denoted by C(p). We had shown that an n-by-n companion matrix can have at most n line segments on the boundary of its numerical range. In this case, this companion matrix is unitarily reducible. In this project, we want to show that if A is an n x n companion matrix, then dW (A) has n — 1 line segments if and only if n is odd and a(A) = {a,au2,... ,aun~1} U {u/a, ^3/a,... ,un~2/a} for some a G C with |a| nearing 1. In this case, A is unitarily reducible. Consequently, if A is an n x n unitarily irreducible companion matrix, then dW (A) has at most n — 2 line segments. |
