Perron's Theorem

If $\boldsymbol{\mu}=(\mu_1, \mu_2, \ldots, \mu_n)$ is an arbitrary set of Positive numbers, then all Eigenvalues $\lambda$ of the $n\times n$ Matrix ${\hbox{\sf A}}=a_{ij}$ lie on the Disk $\vert z\vert\leq M_\mu$, where

$$M_\mu=\max_{1\leq i\leq n} \sum_{j=1}^n {\mu_j\over\mu_i} \vert a_{ij}\vert.$$

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1121, 1979.