PERRON FROBENIUS THEOREM

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1 PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, () each row has some nonzero entry (t s possble to make a transton to each of the th -states from some other state), and (v) some column has more than one nonzero entry (from one of the th -states there are two possble followng states). The entry m s the probablty that somethng taken from the th -state s returned to the th -state. In probablty, usually the rows are assumed to sum to one rather than the columns. However, n that stuaton, the columns are wrtten as rows and the matrx s multpled on the rght of the vector. We retan column vectors whch are multpled on the left by the matrx. A stochastc matrx M s called regular or eventually postve provded there s a q 0 > 0 such that M q 0 has all postve entres. Ths means that for ths terate, t s possble to make a transton from any state to any other state. It then follows that M q has all postve entres for q q 0. A regular stochastc matrx automatcally satsfes condtons () and (v) n the defnton of a stochastc matrx. Let x (0) 0 s the amount of the materal at the th -locaton at tme 0. Then m x (0) s the amount of materal from the th -locaton that s returned to the th -locaton at tme. The total amount at the th -locaton at tme s the sum of the materal from all the stes, x () = m x (0). Let x (q) = x (q). x (q) n be the vector of the amount of materal at all the stes. By the above formula, x () = Mx (0) and more generally x (q+) = Mx (q). Date: Aprl 2, 2002.

2 2 R. CLARK ROBINSON Notce that the total amount of materal at tme s the same as at tme 0: x () = m x (0) = ( ) m x (0) Call ths amount X. Then, = p (q) x (0). = x(q) X s the proporton of the materal at the th -ste at tme q. Lettng ( ) p (q) = = X x(q) be the vector of these proportons, p (q) Mp (q) = M x(q) X = X Mx(q) = X x(q+) = p (q+) also transforms by multplcaton by the matrx M. For a stochastc matrx M, s always an egenvalue of the the transpose of M, M T, wth egenvector. : M T m.. =.. m n Snce M T and M have the same egenvalues, M always has as an egenvalue. Before statng the general result, we gve some examples. Example. Let M = Ths has egenvalues, 0.5, and 0.2. (We do not gve the characterstc polynomal, but do derve an egenvector for each of these values.)

3 PERRON FROBENIUS THEOREM 3 For λ =, M I = Thus, v = 3v 3 and v 2 = 6v 3. Snce we want = v + v 2 + v 3 = ( )v 3 = 0v 3, v 3 = 0., and v = For λ 2 = 0.5, M 0.5 I = Thus, 2v = v 3 and 2v 2 = 3v 3, and v 2 = 3. 2 Notce that v 2 + v2 2 + v2 3 = = 0. Ths s always the case for the egenvectors of the other egenvalues.

4 4 R. CLARK ROBINSON For λ 3 = 0.2, Thus, v = v 3 and v 2 = 0, and M 0.2 I = v 3 = 0. Notce that v 3 + v3 2 + v3 3 = + 0 = 0 s true n ths case as well. If the orgnal dstrbuton s gven by p (0) = = , then M q p (0) = ( ) q 3 + ( ) q whch converges to the dstrbuton v = Take any ntal dstrbuton p (0) wth p(0) =. Wrtng p (0) = y v + y 2 v 2 + y 3 v 3, = p (0) ) ) ) = y ( v + y 2 ( v 2 + y 3 ( v 3 Thus, = y () + y 2 (0) + y 3 (0) = y. p (0) = v + y 2 v 2 + y 3 v 3,

5 PERRON FROBENIUS THEOREM 5 and M q p (0) = v + y 2 (0.5) q v 2 + y 3 (0.2) q v 3 converges to v, the egenvector for λ =. Example 2 (Complex Egenvalues). The followng stochastc matrx llustrates the fact that a regular stochastc matrx can have complex egenvalues. Let M = The egenvalues are λ = and 0.4 ± Notce that 0.4 ± 0. 3 = = 0.9 <. Example 3 (Not Regular). An example of a stochastc matrx whch s not regular (nor transtve) s gven by M = , whch has egenvalues λ =,, 0.5, and 0.3. Notce that states and 2 nterchange wth each other and states 3 and 4 nterchange, but there s no nterchange between the par of stes and 2 wth the par of stes 3 and 4. An example of a stochastc matrx whch s transtve but not regular s gven by M = , whch has egenvalues λ =,, and ± 0.5. Here s s possble to get from any ste to any other ste, but startng at ste one, the even terates are always at ether stes 3 or 4 and the odd terates are always at ether stes or 2. Thus there s no one power for whch all the transton probabltes are postve. Thus, M s not regular. The followng theorem summarzes some of the results about regular stochastc matrces whch the above examples llustrated. Theorem 0.. Let M be a regular stochastc matrx. (a) The matrx M has as a egenvalue of multplcty one. The egenvector v can be chosen wth all postve entres and v =. (It must have ether all postve entres or all negatve entres.) (b) All the other egenvalues λ have λ <. If v s the egenvector for λ, then v = 0. (c) If p s any probablty dstrbuton wth =, then p = v + y v. Also, M q p goes to v as q goes to nfnty. =2

6 6 R. CLARK ROBINSON Sketch of the proof. We assume below that all the m > 0, whch can be done by takng a power f necessary. (a) The multplcty of the egenvalue s one. We noted above M T always has as an egenvalues so t s always an egenvalue of M. Agan, to dscuss the multplcty, we look at M T. Assume M T v = v and not all the v are equal. Assume that k s the ndex for whch v k s largest. By scalar multplcaton by f necessary we can take v k postve. Thus, v k = v k v for all and v k > v l for some l. Then, v k = m k v < m k v k = v k. The second strct nequalty uses the fact that all the m > 0,.e., that M s regular. Snce ths shows v k > v k, the contradcton mples that there are no such other vectors,.e., that there can only be one egenvector for the egenvalue. To complete the proof, we would have to consder the case wth only one egenvector but an algebrac multplcty of the characterstc equaton. We leave ths detal to the references. (b) Case (): Assume λ s a real egenvalue. We show that λ <. Agan, assume that M T v = λv. Let k be such that v k = v k v for all and v k > v l for some l. Then, λv k = m k v < m k v k = v k. Ths shows that λv k < v k, so λ <. Case (): Assume λ s a real egenvalue. We show that λ >. Agan, assume that M T v = λv. Let k be such that v k = v k v for all and v k > v l for some l,.e., v v k for all and v l > v k for some l. Then, λv k = > m k v m k ( v k ) = v k. Ths shows that λv k > v k, so λ >.

7 PERRON FROBENIUS THEOREM 7 Care () Assume λ = r e 2πω s a complex egenvalue wth complex egenvector v. Assume the v are chosen wth v k real and v k Re(v ) for all. Then, Re(λ q v k ) = Re(r q e 2πqω v k ) = Re((M q v) k ) = (M q v)re(v) = ( ) m (q) k Re(v ) < ( ) m (q) k v k = v k. Therefore, r q Re(e 2πqω ) < for all q. Snce we can fnd a q for whch Re(e 2πqω ) s very close to, we need r q < so r = λ <. (c) Let p be a probablty dstrbuton wth p =. The egenvectors are a bass, so there exst y,... y n such that p = y v. Then, = = p Thus, = y v + = y + = y. y =2 y (0) =2 p = v + y v =2 as clamed. Wrtng the teraton as f all the egenvalues are real, M q p = M q v + v y M q v = v + =2 y λ q v =2 whch tends to v because all the λ q < for 2.

8 8 R. CLARK ROBINSON References. Gantmacher, F.R., The Theory of Matrces, Volume I, II, Chelsea Publ. Co., New York, Strang, G., Lnear Algebra and Its Applcatons, Thrd Edton, Harcourt Brace Jovanovch Publshers, San Dego, 988. Department of Mathematcs, Northwestern Unversty, Evanston IL E-mal address: