## Perron-Frobenius Theorem

If all elements of an Irreducible Matrix are Nonnegative, then is an Eigenvalue of and all the Eigenvalues of lie on the Disk

where, if is a set of Nonnegative numbers (which are not all zero),

and . Furthermore, if has exactly Eigenvalues on the Circle , then the set of all its Eigenvalues is invariant under rotations by about the Origin.