Tight bounds on the spectral radius of asymmetric nonnegative matrices (Q1108348)

This paper concerns spectral radius bounds for an arbitrary asymmetric nonnegative \(n\times n\) matrix A. By introducing two concepts of the elementwise product and quotient of two matrices A and B, denoted by \(A*B,((A*B)_{ij}=a_{ij}b_{ij})\) and \(A\div B\), \(((A\div B)_{ij}=a_{ij}/b_{ij})\) where \(b_{ij}\neq 0\) for all i, j in the later case, the author defines two symmetric matrices associated with the matrix A, that is the matrix \(S=(A+A^ T)\), where \(A^ T\) denotes the transpose of A, and the matrix G which is the unique nonnegative matrix such that \(G*G=A*A^ T.\) By means of the graph theoretical point of view, the author associates the nonnegative matrix A with a weighted directed graph D and then shows that if A is any nonnegative matrix and \(G*G=A*A^ T\) then the spectral radius \(\rho\) (A) of A is bounded by \(\rho\) (G)\(\leq \rho (A)\leq \rho (S)\); moreover if D has at least one strong component with maximum spectral radius that happen to be balanced then \(\rho (A)=\rho (G)\) and if A and \(A^ T\) possess a common positive eigenvector with common eigenvalue \(\rho\) (A) then \(\rho (S)=\rho (A)\). Finally three lemmas are given to show whenever a matrix A is balanced.