On Wielandt type inequalities for powers of complex matrices (Q2361680)

Let \(A\) be a complex irreducible \(n\)-square matrix, let \(\rho\) denote the spectral radius, and let \(|\cdot|\) be understood entrywise. \textit{H. Wielandt} [Math. Z.~52, 642--648 (1950; Zbl 0035.29101)] gave a necessary and sufficient condition for (1) \(\rho(A)=\rho(|A|)\). The present authors develop this topic further. First, they prove that (1) holds if and only if \(|A^k|=|A|^k\) for all \(k=1,2,\dots\). Their main result is that it is enough to check (2) \(k=n^2-2n+2\) and this value cannot be decreased. Assume now that \(A\) is real and (entrywise) nonnegative. If \(A\) is primitive, then its exponent~\(\kappa(A)\) is defined as the smallest \(k\) for which \(A^k\) is positive. An interesting connection with (2) is that (3) \(\kappa(A)\leq n^2-2n+2\). Wielandt [loc. cit.], stated (3) without proof and showed that it is sharp. Afterwards, several proofs have been given. Wielandt's original proof, found from his diaries, was published by \textit{H. Schneider} [Linear Algebra Appl. 353, No. 1--3, 5--10 (2002; Zbl 1006.15023)]. If \(A\) is not primitive, then \(\kappa(A)\) is defined as the smallest \(k\) for which all nonzero blocks of \(A^k\) are positive; here \(A\) is in the Frobenius normal form. The present authors prove that (3) holds also in this case. They apply a result of \textit{Š. Schwarz} [Czech. Math. J. 20(95), 703--714 (1970; Zbl 0226.20061)].