Constant norms and numerical radii of matrix powers

Hwa Long Gau, Kuo Zhong Wang, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


For an n-by-n complex matrix A, we consider conditions on A for which the operator norms ||Ak|| (resp., numerical radii w(Ak)), k ≥ 1, of powers of A are constant. Among other results, we show that the existence of a unit vector x in Cn satisfying |〈Ak x,x〉| = w(Ak)=w(A) for 1 ≤ k ≤ 4 is equivalent to the unitary similarity of A to a direct sum (Formula Presented), where |λ| = 1, B is idempotent, and C satisfies w(Ck) ≤ w(B) for 1 ≤ k ≤ 4. This is no longer the case for the norm: there is a 3-by-3 matrix A with ||Ak x|| = ||Ak|| =√2 for some unit vector x and for all k ≥ 1, but without any nontrivial direct summand. Nor is it true for constant numerical radii without a common attaining vector. If A is invertible, then the constancy of ||Ak || (resp., w(Ak)) for k = ±1,±2,… is equivalent to A being unitary. This is not true for invertible operators on an infinite-dimensional Hilbert space.

Original languageEnglish
Article numberOaM-13-72
Pages (from-to)1035-1054
Number of pages20
JournalOperators and Matrices
Issue number4
StatePublished - 2019


  • Idempotent matrix
  • Irreducible matrix
  • Numerical radius
  • Operator norm


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