New bounds for Perron root of a nonnegative matrix (Q874354)

Let \(A\) be a nonnegative \(n\times n\) matrix, \(r\) be the Perron root of \(A\), which is the real eigenvalue of \(A\) of largest in absolute values of all eigenvalues of \(A\). It is shown in this paper that if \(f\) is a polynomial such that \(f(r)\neq0\) and the matrix \(B=f(A)\) is nonnegative, all row sums are nonzero, then for integers \(k\geq0\) and \(m\geq1\) \[ \min_i(\frac{r_i(A^mB^k)}{r_i(B^k)})^{1/m}\leq r\leq\max_i(\frac{r_i(A^mB^k)}{r_i(B^k)})^{1/m}, \] where \(r_i\) denotes the row sum of the \(i\)th row of the matrix so indicated. The inequalities remain valid if the row sums \(r_i\) are replaced by the column sums \(c_i\). Known bounds for the Perron root by \textit{G. Frobenius} [Berl. Ber. 1912, 456--477 (1912; JFM 43.0204.09)] and others may be derived by choosing \(m=1\) and \(B=A^2\).