Application of the coefficient of ergodicity in the estimation of the spectral radius of real matrices

It is shown that every real matrix $A$ can be put in correspondence with a certain stochastic matrix $P$ in such a way that the coefficient of ergodicity $\alpha(P)$ of the matrix $P$ enables us to give an estimate of the spectral radius of the matrix $A$. This estimate takes into account the signs of the elements of $A$, which makes it in many cases more accurate than the generally known estimates. In the case where one of the characteristic values of the matrix $A$ and the characteristic vector corresponding to it are known, an estimate of the localization of the remaining characteristic values of the matrix $A$ is obtained.