Higher-Rank Numerical Ranges of 4-by-4 Matrices

• Main Authors: Yu-Jhau Peng, 彭煜釗
• Other Authors: Hwa-Long Gau
• Format: Others
• Language: en_US
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/539255

碩士 === 國立中央大學 === 數學研究所 === 97 === Let $A$ be an $n$-by-$n$ matrix. For $1leq k leq n$, the rank-$k$ numerical range of $A$ is defined and denoted by $Lambda_k(A) = {lambdainmathbb{C}: PAP=lambda P mbox{ for some rank-{it k} orthogonal projection $P$}}$. In this thesis, we give a complete description of the higher-rank numerical ranges of $4$-by-$4$ matrices. We classify the rank-$2$ numerical ranges of $4$-by-$4$ matrices. Our classification is based on the factorability of the associated polynomial $p_A(x,y,z)equiv mathrm{det}(xmathrm{Re,}A + ymathrm{Im,}A + zI_4)$ of a $4$-by-$4$ matrix $A$. For each class, we also completely determine the shape of the rank-$2$ numerical range of a $4$-by-$4$ matrix.