### Abstract

For an n-by-n matrix A, we determine when the numerical ranges W(A^{k}), k≥1, of powers of A are all equal to each other. More precisely, we show that this is the case if and only if A is unitarily similar to a direct sum B⊕C, where B is idempotent and C satisfies W(C^{k})⊆W(B)for all k≥1. We then consider, for each n≥1, the smallest integer k_{n} for which every n-by-n matrix A with W(A)=W(A^{k})for all k, 1≤k≤k_{n}, has an idempotent direct summand. For each n≥1, let p_{n} be the largest prime less than or equal to n+1. We show that (1)k_{n}≥p_{n} for all n, (2)if A is normal of size n, then W(A)=W(A^{k})for all k, 1≤k≤p_{n}, implies A having an idempotent summand, and (3)k_{1}=2 and k_{2}=k_{3}=3. These lead us to ask whether k_{n}=p_{n} holds for all n≥1.

Original language | English |
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Pages (from-to) | 95-110 |

Number of pages | 16 |

Journal | Linear Algebra and Its Applications |

Volume | 578 |

DOIs | |

State | Published - 1 Oct 2019 |

### Keywords

- Idempotent matrix
- Normal matrix
- Numerical range

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## Cite this

*Linear Algebra and Its Applications*,

*578*, 95-110. https://doi.org/10.1016/j.laa.2019.05.013