# Equality of numerical ranges of matrix powers

Hwa Long Gau*, Kuo-Zhong Wang, Pei Yuan Wu

*Corresponding author for this work

Research output: Contribution to journalArticle

1 Scopus citations

### Abstract

For an n-by-n matrix A, we determine when the numerical ranges W(Ak), 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(Ck)⊆W(B)for all k≥1. We then consider, for each n≥1, the smallest integer kn for which every n-by-n matrix A with W(A)=W(Ak)for all k, 1≤k≤kn, has an idempotent direct summand. For each n≥1, let pn be the largest prime less than or equal to n+1. We show that (1)kn≥pn for all n, (2)if A is normal of size n, then W(A)=W(Ak)for all k, 1≤k≤pn, implies A having an idempotent summand, and (3)k1=2 and k2=k3=3. These lead us to ask whether kn=pn holds for all n≥1.

Original language English 95-110 16 Linear Algebra and Its Applications 578 https://doi.org/10.1016/j.laa.2019.05.013 Published - 1 Oct 2019

### Keywords

• Idempotent matrix
• Normal matrix
• Numerical range