## Abstract

For any n-by-n matrix A, we consider the maximum number k = k(A) for which there is a k-by-k compression of A with all its diagonal entries in the boundary ∂W(A) of the numerical range W(A) of A. For any such compression, we give a standard model under unitary equivalence for A. This is then applied to determine the value of k(A) for A of size 3 in terms of the shape of W(A). When A is a matrix of the form 0 ^{w1}0wn- _{1} ^{wn}0, we show that k(A)=n if and only if either | ^{w1}|=⋯=| ^{wn}| or n is even and | ^{w1}|=| ^{w3}|=⋯=|wn- _{1}| and | ^{w2}|=| ^{w4}|=⋯=| ^{wn}|. For such matrices A with exactly one of the ^{wj}'s zero, we show that any k, 2≤k≤n-1, can be realized as the value of k(A), and determine exactly when the equality k(A)=n-1 holds.

Original language | English |
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Pages (from-to) | 514-532 |

Number of pages | 19 |

Journal | Linear Algebra and Its Applications |

Volume | 438 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2013 |

## Keywords

- Compression
- Numerical ranges
- Weighted shift matrix