## Abstract

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^{-1}_{n}) 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 S_{n-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^{-1}_{n} -matrices. Finally, we consider the numerical ranges of S^{-1}_{n} -matrices and S_{n} -matrices, and give a generalization of a result of Chien and Nakazato on tridiagonal matrices (cf. [3, Theorem 7]).

Original language | English |
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Pages (from-to) | 865 |

Number of pages | 1 |

Journal | Operators and Matrices |

Volume | 7 |

Issue number | 4 |

DOIs | |

State | Published - 2013 |

## Keywords

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