| Summary: | 碩士 === 東吳大學 === 數學系 === 96 === Let A be an n-by-n complex matrix. The numerical range of A is theset W(A) = {x*Ax : x belongs to Cn, x*x = 1}. We deal with the circularityproperty of the numerical range of certain nilpotent upper triangularToeplitz matrices. If A is the nilpotent upper triangular Toeplitz matrix of the forms A(a1,a2,0)belongs to M4 or A(a1,a2,0,0)belongs to M5, we provethat W(A) is a circular disk centered at the origin if and only if a1a2=0.Similarly, if A is the nilpotent upper triangular Toeplitz matrix of theforms A(a1,0,a3)belongs to M4 or A(a1,0,a3,0)belongs to M5, W(A) is a circular disk centered at the origin if and only if a1a3=0. If A is the nilpotentupper triangular Toeplitz matrix of the forms A(a1,0,0,a4)belongs to M5 orA(a1,0,0,a4,0)belongs to M6, we prove that W(A) is a circular disk centeredat the origin if and only if a1a4=0. Furthermore, if A is the nilpotent
upper triangular Toeplitz matrix of the forms A(a1,0,a3)belongs to M4 orA(a1,0,0,a4)belongs to M5, A(a1,0, .., 0,an−1)belongs to Mn, we prove that W(A) isa circular disk centered at the origin if and only if a1an−1=0. Finally,if A is the nilpotent upper triangular Toeplitz matrix of the forms
A(0, 0, .., 0, a[n/2], a[n/2]+1, .., an−1)belongs to Mn, we prove that W(A) is a circular
disk centered at the origin if and only if an/2a(n/2)+1...an−1=\= 0.
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