Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices i the real field have ad the exam if they hold i the fiite fields. We will prove how, give the coditios of a fiite field, properties hold or fail to hold. We will exted our scope to 3 3 stochastic matrices ad lay the groudwork for future research. Fially, we show how the properties i the real field exted directly to aother ifiite field: the ratioals. Itroductio Stochastic processes ca be used to model may pheomea ivolvig evets (states) that evolve over time. This pheomea ca be the behavior of the price of a stock give a set of time periods or weather patters. I precise terms, a stochastic process is a, idexed collectio of radom variables (Hillier) where the idex is usually the oegative itegers. We arrow our focus to stochastic matrices, which are particularly useful i defiig trasitio probabilities. Kowig the properties of such matrices allows to defie solutios to problems that ivolve their use, amog other thigs. Its applicatio aside, however, we shall study their behavior i fiite fields. Literature review ad Properties of Stochastic Matrices i A stochastic matrix is a square matrix such that for all of its etries, p ij,0 p ij, ad that p ij, for all j (a colum stochastic matrix), or p ij for all i ( a row stochastic matrix), or j j i ad j, respectively (a doubly stochastic matrix). Property i p ij ad p ij for all i i Cosider a row stochastic matrix, P, ad its eigevalues. It is clear that Pe e T where e. Hece,,e is always a eigepair of a row stochastic matrix. I the case of the colum stochastic matrix, say, P T, we similarly have e T P T e T. For a doubly stochastic matrix P, both Pe e ad e T P T e T hold. Property 2 Cosider the -orm ad the -orm of the three types of stochastic matrices. Recall that the -orm of a matrix, A a ij, is defied as A max j
i a ij ad its -orm is defied as A max i i a ij. Simply, put, the former is the maximum colum sum ad the latter is the maximum row sum. The -orm for a row stochastic matrix P is hece equal to oe ad its -orm, P : P. It is clear why for the case of the -orm the value is less tha. To see why the lower limit is, cosider the fact that the sum of all the etries is so if all colum sums are less tha, the the sum of all the colum sums is less tha : a cotradictio. For a colum stochastic matrix P, sicep T is a row stochastic matrix, we have P ad P. Cosequetly, for a doubly stochastic matrix, P, P P. Property 3 The spectral radius of a matrix A is defied as A max i,..., i where i for i,2... are eigevalues of A. We kow that,e will always exist as a eigepair for ay stochastic matrix. Now, the absolute value of ay eigevalue caot be greater tha ay orm of the matrix to which it is a eigevalue. Hece, the spectral radius of ay stochastic matrix is sice we kow that, e is a eigepair of ay stochastic matrix ad ay stochastic matrix either has a orm or -orm that equals. Property 4 It is kow that for ay square matrix, the sum of our stochastic matrices eigevalues equal its trace ad the product of their eigevalues equal their determiats. We prove these two results for 2 2 matrices as follows. Let A a c b d. The the characteristic polyomial of A is det A I a d bc 2 a d ad bc 0. () Let ad 2 be eigevalues of A. Because ad 2 are roots of det A I 0, det A I 2 2 2 2 0. (2) Comparig coefficiets of ad costat terms i () ad (2), we obtai 2 a d trace A, ad 2 ad bc det A. Property 5 For a 2 2 row stochastic matrix, if all of its etries are strictly greater tha 0 the eigevalue issimple. a a Let P bea2 2row stochastic matrix. The P where b b 0 a,b. Let 2 be aother eigevalue of P. Because 2 a b, 2 a b. Sice 0 a,b, a b 2. 2
Property 6 Observe from Property 5 that 2 0 if ad oly if a b, whe a b P. Hece, for a 2 2 row stochastic matrix whose rows are ot liearly a b idepedet, 2 0. Let P bea2 2doubly stochastic matrix. The a a P a a 2a ora. ad 2 2a. is a simple eigevalue if ad oly if Empirical Work: All of these properties rely o a crucial assumptio: all the etries are real umbers. Withi the scope of elemetary mathematical methods, we shall ow explore the properties of stochastic matrices with etries i fiite fields. We first restrict our aalysis to 2 2 matrices with etries i Z p,wherep is a prime umber. We will also establish that property holds i ay fiite field Z p as well.. Row Stochastic Matrices i Z 2 : There are oly 2 2 2 2 6 possible matrices of which eight are either row, colum or doubly stochastic. we shall restrict our attetio to row stochastic matrices, ievitably a superset of doubly stochastic matrices. Matrices 0 0 0 0 P, P 2, P 3 ad P 4 are all such 0 0 0 0 row stochastic matrices. Because the first three matrices are either lower or upper triagular matrices, their eigevalues are their diagoal elemets which are or 0. Their eigevectors ca be obtaied by solvig the system of liear equatios: A I X 0. Hece, eigepairs of P, P 2 ad P 3 are: 0 P :,,, ; P 2 :,, 0, 0 ;ad P 3 :,, 0, 0. x Eigevalue 2 of P 4 satisfies the equatio: 2 0 2. Let v y be a eigevector correspodig to eigevalue. The the equatio P 4 v v implies that 3
P 4 v y x x y,thatis,x y. Hece, P 4 has oly oe eigepair:,. Clearly,Property:,e is a eigepair of P, ad Property 3: P, hold for row stochastic matrices i Z 2. Sice i Z 2, there is o row stochastic matrix with all o-zero etries, Property 5 caot be applied. 2. Row Stochastic Matrices i Z 3 : I Z 3, we have a total of 8 matrices to begi with. Of the matrices, we will agai cosider oly the row stochastic matrices. The rows a,b ad c,d of such matrices should hece be oe of the pairs of values,0 or 2,2 ad we have five such matrices, which iclude P i, i,2,3,4 that we aalyzed i Z 2. The additioal five row stochastic matrices are: P 5 0, P 6 0, P 7 0, P 8 0, P 9. Eigevalues of P 5 ad P 8 (they are lower ad upper triagular matrices, respectively) are 0 ad 2. So, Property 3 o loger holds. Let us revisit P 4.I 0 Z 3, 2 0 3 2. P 4 also has eigevalues ad 2. Cosider the stochastic matrix whose etries are all strictly ozero i Z 3 : P 9. I this case, 2 4 3 3 0. So, both Property 3 ad Property 5 hold. 3. Row Stochastic Matrices i Z 5 : For Z 5, we will deal with 525 possible matrices. Clearly, the row stochastic matrices must have i their row etries a,b, c,d should be oe of the pairs 3,3, 4,2, or,0. Through simple computatio, we have 25 row stochastic matrices. Rather tha calculatig the eigepairs of every such matrix, to verify whether the properties hold for Z 5, we shall first try to fid simple couterexamples. I Z 5, we ca ideed fid a couterexample to Property 3: 4 2 2 4 for 2 4 4 7 5 2. 3 3 The couterexample to Property 5: for 2 3 4 6 5. 2 4 We ca therefore coclude that while Property holds true i ay fiite field Z p, Properties 3 ad Property 4 do ot always hold. I the followig, we shall aalyze why ideed certai properties hold. Geeral Results: 4
Row stochastic matrices i the fiite fieldz p where p is a prime umber have a a a eigevalue of 0 if ad oly if det 0. While the case i the real field is b b a a trivial, the oly matrix beig of the form, this will hold true i Z p as a a log as the rows are liearly depedet. Let a, a ad b, b Z p Z p such that there exists a k where a, a k b, b so ak bad k a b. The a a det a b b a a ab b ba a b so a b ad like b b a a i the real case we have the same form as we do i the real case:. a a Notice that all the eigevalues obtaied i the examples above have solutios i Z p. We will prove that all row stochastic matrices have such a property. Give the row a a stochastic matrix P, our characteristic equatio will be b b a b a b 0. We kow that, is a solutio ad it is clear that 2 a b Z p. Hece all eigevalues for 2 2 row stochastic matrices exist withi the fiite field to which the stochastic matrix is restricted. We ca see that property 6 holds here as well: a b a b tr P ad a b det P ab a b ab a b ab a b. a a Doubly stochastic matrices will also have the form,which a a will yield the characteristic equatio a 2 a 2 0adhere,asitheldithe real case,,2,2a ad, agai, is simple if ad oly if a 0. Ulike stochastic matrices i the real field, it is possible to have a eigevalue greater tha i the fiite field. We kow this from oe of the examples above. We also kow the solutios to the characteristic equatios of ay stochastic matrix belog to the fiite field ad hece, o field extesios are ecessary to have solutios. 3 3 Stochastic Matrices i Z 2 ad Z 3 We ca exted our scope to 3 3 stochastic matrices ad see how the properties we have derived applies to them. It is still obvious that,e is a eigepair to this 5
matrix: a b c d e f g h i sice a b c d e f g h i. Cosiderig row stochastic matrices i Z 2, the oly possible row etries we have are,0,0 ad,,. ForZ 3,wehave 2,,, 2,2,0,,0,0. Let us look ito their eigevalues. We kow that if the solutios to their characteristic equatios are all withi their fields (without extesios), we will have a characteristic equatios of the form 2 3.We kow that if det a b c d e f g h i a b c d e f g h i is liearly depedet, 0 ad like i the 2 2 case, we have a eigevalue equal to 0 ad the other must be i Z 3. While it would be more challegig to fid the geeral form i which 3 3 matrices would have simple or distict eigevalues we ca still apply some of the properties from above. For istace, the 3 3 idetity matrix clearly has a ozero determiat ad its characteristic equatio is ( 3 0. This holds true for both Z 2 ad Z 3. Liear idepedece is hece a ecessary but ot sufficiet coditio for the eigevalue of to be simple. Stochastic Matrices i the Field of Ratioal Numbers It is obvious that Properties, 2, 3 hold i the field of ratioals. Ideed all properties that held i the real field hold also i the field of ratioals as well. However, we shall explore their characteristic equatio ad solutios ad fid that they take o well-defied forms. Cosider row stochastic matrix p,q,r ad s are itegers. The a a b b 2 a b p q r s where a p q ad b r s,ad sp rq qs qs is agai a ratioal umber. For a doubly stochastic matrix a p q,adp,q are itegers. The a a a a where 2 2a 2p q 2p q qs is agai a ratioal umber. 6
Coclusio I additio to verifyig how some properties that held for stochastic matrices i the field of real umbers held i the fiite field as well, we have proved aalytically why i fact they did or did ot hold. Our mai result beig the property that 2 2 stochastic matrices will have solutios to their characteristic equatios that are i their respective fiite fields ad as a cosequece, the trace ad determiat of ay stochastic matrix with a fiite field equal the sum ad product of its eigevalues respectively. We have also prove that we have oly oe trivial case for a liearly depedet row stochastic matrix i the fiite field as well. O the other had, we have proved, maily through couterexample, that some properties that held i the real field did ot hold i the fiite field. We have foud that the spectral radius is ot less tha or equal to ad that stochastic matrices whose etries are all strcictly positive do ot ecessarily have as a simple eigevalue. We have also proved why the properties we have derived as we explored such matrices i the real field hold i a ifiite subfield (the ratioal umbers) of the real field ad have laid the foudatio for future work by extedig our scope to 3 3 stochastic matrices. Refereces [] Leo, Steve, Liear Algebra, 7th Editio, Pretice Hall, Upper Saddle River, New Jersey, 2006 [2] Stewart, William, Itroductio to the Numerical Solutios of Markov Chais, Priceto Uiversity Press, Priceto, New Jersey,994 [3] Hillier, Frederick, Itroductio to Operatios Research, McGraw Hill, New York, New York, 2005 7