Matrices with defect index one

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]).