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),
Furthermore, if 
has exactly 
eigenvalues 
on the circle 
, then the set of all its eigenvalues
is invariant under rotations by 
about the origin.
See also
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References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1121, 2000.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Perron-Frobenius Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Perron-FrobeniusTheorem.html