Matrices with defect index one (Q2873572)

The defect index of an \(n\times n\) complex matrix \(A\) is the rank of \(I_n-A^*A\). It is a way to measure how far \(A\) is from the unitary matrices. An \(n \times n\) matrix \(A\) of defect index one is said to be of class \(\mathcal{S}_n\) (resp., \(\mathcal{S}_n^{-1}\)) if its eigenvalues are all in the open unit disc (resp., in the complement of closed unit disc). In the paper under review, the authors give some characterizations of matrices which have defect index one. In particular, they show that an \(n\times n\) matrix A is of defect index one if and only if A is unitarily equivalent to \(U\oplus C\), where \(U\) is a \(k\times k\) unitary matrix, \(0\leq k<n\), and \(C\) is either of class \(\mathcal{S}_{n-k}\) or of class \(\mathcal{S}_{n-k}^{-1}\). They also give a complete characterization of polar decompositions, norms and defect indices of powers of \(\mathcal{S}_n^{-1}\)-matrices. In addition, they investigate the numerical ranges of \(\mathcal{S}_n^{-1}\)-matrices and \(\mathcal{S}_n\)-matrices, and give a generalization of a result on tridiagonal matrices of \textit{M.-T. Chien} and \textit{H. Nakazato} [J. Math. Anal. Appl. 373, No. 1, 297--304 (2011; Zbl 1205.47007)].