Matrices with defect index one

Cheng Chun Chang, Hwa Long Gau, Ya Shu Wang, Szu Chieh Wu, Ya Ting Yeh

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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 Sn (resp., S-1n) 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 ⊕C, where U is a k -by- k unitary matrix, 0 ≤ k < n, and C is either of class Sn-k or of class S-1 n-k. We also give a complete characterization of polar decompositions, norms and defect indices of powers of S-1n -matrices. Finally, we consider the numerical ranges of S-1n -matrices and Sn -matrices, and give a generalization of a result of Chien and Nakazato on tridiagonal matrices (cf. [3, Theorem 7]).

Original languageEnglish
Pages (from-to)865
Number of pages1
JournalOperators and Matrices
Issue number4
StatePublished - 2013


  • Defect index
  • Numerical range
  • Polar decomposition
  • S -matrix
  • S -matrix


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