Perron-Frobenius Theorem

If all elements a_(ij) of an irreducible matrix A are nonnegative, then R=minM_lambda is an eigenvalue of A and all the eigenvalues of A lie on the disk


where, if lambda=(lambda_1,lambda_2,...,lambda_n) is a set of nonnegative numbers (which are not all zero),


Furthermore, if A has exactly p eigenvalues (p<=n) on the circle |z|=R, then the set of all its eigenvalues is invariant under rotations by 2pi/p about the origin.

See also

Wielandt's Theorem

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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

Perron-Frobenius Theorem

Cite this as:

Weisstein, Eric W. "Perron-Frobenius Theorem." From MathWorld--A Wolfram Web Resource.

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