Improvement of Frobenius bounds for the Perron root of a non-negative matrix
According to the Perron-Frobenius theorem, the spectral radius \(r(A)\) of a nonnegative matrix \(A\) is its eigenvalue. The number \(r(A)\) is referred to as the Perron root of \(A\). Frobenius established that \(s\leq r(A)\leq S\), where \(s\) and \(S\) are the least and the greatest row sums of the elements of \(A\). This paper presents a method for constructing second-order matrices whose Perron roots are the upper and lower bounds for the Perron root of a given nonnegative matrix of arbitrary order. The bounds for the Perron root that are obtained by this method are shown to be better than both Frobenius bounds and those obtained by subsequent investigators.
scientific article; zbMATH DE number 1440448