Bounds for the Perron root of a nonnegative matrix involving the properties of its graph
New bounds for the Perron root of a nonnegative matrix are given. By the Perron-Frobenius theorem, the spectral radius of a nonnegative matrix is one of its eigenvalues. This eigenvalue is called the Perron root of the matrix. Frobenius gave bounds for the Perron root in terms of the minimal sum and the maximal sum of the elements of the matrix over rows. The bounds given here involve properties of the digraph associated with the matrix. Let \(s_i\) be the sum of elements of the \(i\)th row. The weight of a path \(i_1, \dots, i_k\), \(i_{k + 1}\) in the graph of the matrix is the product \(s_{i_1} \dots s_{i_k}\) and the mean weight of the path is the geometric mean \((s_{i_1} \dots s_{i_k})^{1/k}\). The main result is: if \(A\) is a nonnegative matrix without zero rows, \(\nu\) the minimal and \(w\) the maximal mean weight of simple contours of its graph, then the Perron root \(r\) satisfies the inequalities \(\nu \leq r \leq w\). If \(A\) is indecomposable, then \(\nu < r < w\).
scientific article; zbMATH DE number 898968