Numerical Radii of Matrices and its Submatrices

• Main Authors: Zhen-Yu Ye, 葉鎮宇
• Other Authors: Hwa-Long Gau
• Format: Others
• Language: en_US
• Published: 2018
• Online Access: http://ndltd.ncl.edu.tw/handle/884j94

碩士 === 國立中央大學 === 數學系 === 106 === Let A=[a_ij]_(i,j=1)^n and A' be a weighted shift matrix of weights {a_i,i+1} for all i=1,...,n We know that w(A)≥w(A') by Proposition 3.1{1}. In this thesis, we consider when the equality w(A)=w(A') holds. In this thesis, we obtain some classes of matrices A for which w(A)=w(A') implies A=A'. We show that (1) if A is a non-negative matrix, then w(A)=w(A') if and only if A=A', (2) if A is a Toeplitz matrix, then w(A)=w(A') if and only if A=A', and (3) if A is a circulant matrix, then w(A)=w(A') if and only if A=A'. Note that A' is a weighted shift. We also consider when the equality w(A)=w(A') holds if A' has periodic nonzero weights. We first study the period of weights of A' is one. The sufficient and necessary condition of the equality w(A)=w(A') is given. Next, we concerned with the period of weights of A' is even. We show that if w(A)=w(A'), then A is unitarily reducible. Finally, the case that the period of weights of A' is odd is also considered in this thesis.