Perron-Frobenius Theorem

If all elements $a_{ij}$ of an Irreducible Matrix ${\hbox{\sf A}}$ are Nonnegative, then $R=\min M_\lambda$ is an Eigenvalue of ${\hbox{\sf A}}$ and all the Eigenvalues of ${\hbox{\sf A}}$ lie on the Disk

$$\vert z\vert\leq R,$$

where, if $\boldsymbol{\lambda}=(\lambda_1, \lambda_2, \ldots, \lambda_n)$ is a set of Nonnegative numbers (which are not all zero),

$$M_\lambda=\inf\left\{{\mu: \mu\lambda_i>\sum_{j=1}^n \vert a_{ij}\vert\lambda_j, 1\leq i\leq n}\right\}$$

and $R=\min M_\lambda$. Furthermore, if ${\hbox{\sf A}}$ has exactly $p$ Eigenvalues $(p\leq n)$ on the Circle $\vert z\vert=R$, then the set of all its Eigenvalues is invariant under rotations by $2\pi/p$ about the Origin.

See also Wielandt'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.