# how to proof the sabinvariance theorem?

the sabinvariance theorem:
Let T be a non-negative irreducible matrix,s a positve number.and y>=.=0.a vector satisfying Ty<=sy.
Then (a) y>0;(b)s>=r,where r is the Perron-Frobenius eigenvalue of T.
Moreover,s=r if and only if Ty=sy.