# Perron Frobenius Theorem modified

On this site I found a modified version of Perron Frobenius Theorem

Perron-Frobenius Theorem: If M is a positive, column stochastic matrix, then:

1 is an eigenvalue of multiplicity one.

1 is the largest eigenvalue: all the other eigenvalues have absolute value smaller than 1.

the eigenvectors corresponding to the eigenvalue 1 have either only positive entries or only negative entries. In particular, for the eigenvalue 1 there exists a unique eigenvector with the sum of its entries equal to 1.

In this same site, in Disconnected components section there is a matrix

$$\begin{matrix} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1/2 & 1/2 \\ 0 & 0 & 1/2 & 0 & 1/2 \\ 0 & 0 & 1/2 & 1/2 & 0 \\ \end{matrix}$$

This matrix satisfies the conditions therefore for the eigenvalue 1 there exists a unique eigenvector with the sum of its entries equal to 1.

but at least exists two vectors

$$u = (1/2,1/2,0,0,0)$$

$$v = (0,0,1/3,1/3,1/3)$$

what happened? Is modified theorem incorrect?

Thank you!

The $$5\times5$$ matrix is not positive. It has zero entries.