Question

4. (a) Write down, without proof, all parts of the Perron-Frobenius Theorem (b) Let S be a stochastic matrix. Prove that 1 is the Perron eigenvalue of S, and e (1 Furthermore, prove that A-1 for every eigenvalue λ of S 1) is the corresponding Perron eigenvector of S (c) For each of the given matrices S(a) below determine the values of the parameter di for which the limit link oo (Si exists. Justify your answer! 1 1 2 2

Answer 1

(a) t Perron- Frobeniss tuerew for requlor matricer Suppose-ne-Rnxmù non: negative aud. regule, e theu ian iuvalut g of A at is real and 0SI the sigevaliu one--and torre sponds to a 1x1 Tordan block the hft ts calld fhe feron frobenius (PE) Eigen ferron Frobenlus theorem for non- negativn malries: uve is an sa rf thatis rea aud hou-igatie uwith assoriatidu mon-negative teft and rigut sigewvectore hrg is talld fle ferron- Frobendes tr) eiguvalur of A te assotiated won-uegative t left and rigw) ciseueetorsa

4 (b)1 ns-Con be, eauty.dkeon+.w Seen wtich e dinetos the aul- Oues vecter apropriate siza 1 e (S) Now, S is sto chasti hara Hhat above/-vue Nove ;, and tur e.