MATRICES WITH DEFECT INDEX ONE

作者：Chang Cheng Chun^*; Gau Hwa Long; Wang Ya Shu; Wu Szu Chieh; Yeh Ya Ting

来源：Operators and Matrices, 2013, 7(4): 865-885.

DOI：10.7153/oam-07-48

• polar decomposition
• defect index
• numerical range
• S-n-matrix
• S-n(-1)-matrix

In this paper, we give some characterizations of matrices which have defect index one. Recall that an n-by-n matrix A is said to be of class S-n (resp., S-n(-1)) if its eigenvalues are all in the open unit disc (resp., in the complement of closed unit disc) and rank (In - A*A) = 1. We show that an n-by-n matrix A is of defect index one if and only if A is unitarily equivalent to U circle plus C, where U is a k-by-k unitary matrix, 0 %26lt;= k %26lt; n, and C is either of class Sn-k or of class S-n-k(-1). We also give a complete characterization of polar decompositions, norms and defect indices of powers of S-n(-1)-matrices. Finally, we consider the numerical ranges of S-n(-1)-matrices and S-n-matrices, and give a generalization of a result of Chien and Nakazato on tridiagonal matrices (cf. [3, Theorem 7]).

• 出版日期2013-12
• 单位中央民族大学