## Abstract

We derive a matrix model, under unitary similarity, of an n-by-n matrix A such that A,A^{2},⋯,A^{K} (k≥1) are all partial isometries, which generalizes the known fact that if A is a partial isometry, then it is unitarily similar to a matrix of the form [0B 0C] with B* B+C*C=I. Using this model, we show that if A has ascent k and A, A ^{2},⋯,A^{k-1} are partial isometries, then the numerical range W(A) of A is a circular disc centered at the origin if and only if A is unitarily similar to a direct sum of Jordan blocks whose largest size is k. As an application, this yields that, for any S_{n}-matrix A, W(A) (resp., W(⊗ A)) is a circular disc centered at the origin if and only if A is unitarily similar to the Jordan block J_{n}. Finally, examples are given to show that, for a general matrix A, the conditions that W(A) and W(⊗ A) are circular discs at 0 are independent of each other.

Original language | English |
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Pages (from-to) | 325-341 |

Number of pages | 17 |

Journal | Linear Algebra and Its Applications |

Volume | 440 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2014 |

## Keywords

- Jordan block
- Numerical range
- Partial isometry
- Power partial isometry
- S -matrix