Frobenius Theorem

Let ${\hbox{\sf A}}=a_{ij}$ be a Matrix with Positive Coefficients so that $a_{ij}>0$ for all $i,j=1$, 2, ..., $n$, then ${\hbox{\sf A}}$ has a Positive Eigenvalue $\lambda_0$, and all its Eigenvalues lie on the Closed Disk

$$\vert z\vert\leq\lambda_0.$$

See also Closed Disk, Ostrowski's Theorem

References

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