 Advances in Pure Mathematics, 2011, 1, 105-117 doi:10.4236/apm.2011.14023 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Left Eigenvector of a Stochastic Matrix Sylvain Lavalle´e Departement de mathematiques, Universite du Quebec a Montreal, Montreal, Canada E-mail: sylvain.lavallee@uqtr.ca Received January 7, 2011; revised June 7, 2011; accepted June 15, 2011 Abstract We determine the left eigenvector of a stochastic matrix associated to the eigenvalue 1 in the commu- tative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is , where is the M1(,, )nNN iNith principal minor of , where is the identity matrix of dimension . In the noncommutative case, this eigenvector is , where is the sum in of the corresponding labels of nonempty paths starting from and not passing through in the complete directed graph associated to . =nNMI11(,PinIn1),nPiPijaiM Keywords: Generic Stochastic Noncommutative Matrix, Commutative Matrix, Left Eigenvector Associated To The Eigenvalue 1, Skew Field, Automata 1. Introduction It is well known that 1 is one of the eigenvalue of a stochastic matrix (i.e. the sum of the elements of each row is equal to 1) and its associated right eigenvector is the vector . But, what is the left eigenvector associated to 1? (1,1, ,1)TIn the commutative case, this eigenvector is the vector 1(,, )nMM , where iM is the i principal minor of the matrix. This formula is known in probability theory; it amounts to finding the stationary distribution of the finite Markov chain whose transition matrix is the stochastic irreducible matrix. thIn the noncommutative case, we must involve inverses of elements of the skew field and as these may be undefined, we take a generic noncommutative stochastic matrix: this is the matrix of noncommuting vari- ables ij subject only to the stochastic identities; i.e. the sum of each row equals one. ()ijaaWe work in the free field generated by these variables (in the sense of Paul Cohn), which we call the stochastic free field. Considering the complete digraph on the set , let i{1,,}nM be the set of paths from to i. Let i be the paths starting from and not passing through again. We identify i with the noncom- mutative power series which is equal to the sum of all the words in i and we still denote this series by i. Next we show that the elements iP iiPPP1iP can be evaluated in the stochastic free field and that the vector is fixed by our matrix; moreover, the sum of the 11(,,nPP1iP1) is equal to 1, hence they form a kind of noncommutative limiting probability. These results have been proved in  but the proof proposed in this paper are completely different. Indeed for the major part the proof in these two instances, we use elementary operations on the rows and the columns of the matrix. 2. Commutative Case The commutative case is well known in probability theory. Indeed, we calculate the limit probability of a finite Markov chain by replacing the stochastic matrix by MMI, where is the identity matrix of the appropriate dimension. For this, we use the Markov chain tree Theorem, where this calculus is expressed in terms of spanning trees. The Markov chain tree theorem gives a formula for the stationary distribution of a finite Markov chain. Equivalently, this formula gives a row vector fixed by a matrix fixing . This theorem is attributed to Kirchoff by Persi Diaconis, who gives a probabilistic proof of it (see  p. 443 and 444). See also [3,4]. I(1,1, ,1)TThe proof uses only the definition of the determinant which involves the principal minors. Proposition 1 Let be a stochastic matrix and M S. LAVALLE´E Copyright © 2011 SciRes. APM 106 n1,==()nijijbNMIn; i.e. for all , =1,,in=1 =0ijjb. Then , where is the i1(,, )nNN iNth principal minor of , is the left eigenvector associated to the eigenvalue 0. NProof. By definition of , we have Ndet= 0N. We show that 12(, ,, )=0nNN N N We verify this Equation with the first column of . N 111111121, 122 23221222, 132 33311 11,11,21, 123112 1,1=222 232222 2,132 333=211231,=2==nnnnnnnnn nnnn nnnjnjnnjnnjnn nnnnj njNbN bbb bbb bbb bbb bbbbb bbb bbb bbb bbb bbb bbbb bbb  11,21, 112121, 122 23222222, 132 33311 11,21,21, 123=01, 1121,12, 1222,11, 11,21, 1=0=nnnnnnnnnn nnnn nnnnnnnn nnnbbbb bbb bbb bbb bbbbb bbb bbb bbb bbb b   112 1,1222 2,1111,1,21, 1112 1,122 2322222,132 33311 11,1,21, 12322 23232 3331123==(1nnnnnnnn nnnnnnnnnnnn nnnnn nnnnnn nnbb bbb bbbbb bbb bbb bbb bbb bbbbb bbb bbb bbb bbbb b    12 13122 232211,2 1,31,12 13122 23222 23232 333111 11,2 1,31,23)=(1)=det =0nnnnnn nnnnnnnnnn nnnn nnbbbbb bbbb bbb bbb bbb bbb bbbbb bbb b   N Remark 2 If is reducible, then this proposition is asserting that the zero vector ia an eigenvector. If is irreducible and stochastic, then its adjoint will be of the form , where is a null vector. Te pro- MM(1,,1) T uTu S. LAVALLE´E Copyright © 2011 SciRes. APM 107position follows from the cofactor formula for the adjoint. 3. Noncommutative Case In , the authors have prove in two manners that are actually similar, Theorem 9. In the first, results from variable-length prefix codes are required; and the second deals with general variable-length codes, not necessarily prefix. Moreover, in Appendix 2 of , we see how the theory of quasideterminants may be used to obtain these results on noncommutative matrices. The major part of the new proof of this Theorem involves only the elementary operations on the rows and on the columns of the stochastic matrix. Therefore, we need the following results. 3.1. Languages and Series Let A be a finite alphabet and *A be the free monoid generated by A. A language is a subset of a free monoid *A. A language is rational if it is obtained from finite languages by the operations (called rational) union, product (concatenation) and star. The product of two languages is 12LL12 1122,wwwLwL, and the star of is L*10=,0=nni nLwwwLn L . Ra- tional languages may be obtained by using only unam- biguous rational operations; these are: disjoint union, unambiguous product (meaning that if , then has a unique factorization 12, ii12wLLwww=wwL) and the star restricted to languages which are free submonoids of *L*A. A formal series is an element of the -algebra of noncommutative series A , where A is a set of noncommuting variables. A rational series is an element of the smallest subalgebra of A , which contains the -algebra of noncommutative polynomials A, and which is closed under the operation 1*=0==1nnSS SS which is defined if has zero constant term. We denote by the -algebra of rational series. SratA Let be a rational language. Since may be obtained by unambiguous rational expressions, it follows L Lthat its characteristic series wL is ratio- nal. We shall identify a language and its characteristic series. This is exposed in  or . wA3.2. Free Field Let be a field. If the multiplicative group of is commutative, then is a commutative field. Else, is called skew field. The ring of rational formal series in noncommutative variables   ratA  is not a skew field. However, skew fields containing A do exist. One of them is the free field (in the sense of Cohn), denoted . The free field is the noncommutative analogue of the field of fractions of the ring of commu- tative polynomials. There are several constructions of the free field: Amitsur , Bergman , Malcolmson , Cohn  and . A square matrix of order n is full if it can’t expressed as a product of matrices 1nn by . 1nnTheorem 3 (, Thm 4.5.8 .) Each full matrix with coefficients in MX is invertible in the free field. A square matrix of order is called ho llow if it contains a npq block of zeros such that 1pqn. Example 4 The matrix 001001111 is hollow since there is a 22 block of 0 and . 4>3Proposition 5 (, Prop. 4.5.4) A hollow matrix is not full. Proposition 6 Take , and 1nnnM1n, where is a skew field. We suppose that M is invertible. Then 0M is invertible if and only if 10M. Proof. () Suppose that , then the inverse 1=cM0of 0M is 1111111 1 1MMcMMccM c Indeed, this matrix is the right inverse of since M 11111111 11111 11111111 1111 111 11111 11==0()(=()1=cccM        1)  MMM MMccM cMMMcMcMM ccMMcM MccMcMc cMMcM Mc     10=01 S. LAVALLE´E Copyright © 2011 SciRes. APM 108 This matrix is also the left inverse of since M11111111 11 111111 11111 1111111111 1=11=0()()=() ()1cc      MMMcMMccM cMMcMMMc MMcMcMMc cMMcMcMMc McM  11=cc   10=01 Suppose that ()0M is invertible. Let M be its inwe have Hence 1verse. Then 0=00M 1==0M and =00. It follows that 1=M 1==M  1is proposition, we de follory 7 Let 10M From theduce thwing result. Corolla ,M and  be with coefficmatrices ients in A where M is full. If 0M is not full (in paar, if rticul 0M is by elementary operations on the lines and on the columns to a hollow 3.Let be a generic noncommutative matrix; tive varld. Weequivalent matrix) then in the free field. 1=0M 3. Generic Stochastic Matrices i.e. the ija are noncommutaiables. We denote the corresponding free fie associate to M 1,=ij ijnaM the matrix S: it’s exactly the same matrix of which the coefficients satisfy the dentities stochastic i (1) In other words, the sum of each line of =1j=1,, ,=1nijina  S is equal to 1; hence S is a stochastic matrix. We call S a gene- ric noncommutative stochastic matrix. The algebra over generated by its coefficients is associative ra, it is isomorph to the algebralgeb free aa since,ai jij. e stochastic /(1)ijhby Indeedn eliminate the with trelations We denote thbra , we ca (1).iiais algea. field called Hence,e is a corresponding freestochastic free field denoted therS. 3.4. Paths Consider the set of nonempty paths in the complete directed graph with a set of vertices 1,, n starting from i and not passing through i; we denote by iP the sum in ija of all the corresptional series, and tof the leonding words. It is defines an element classically a rahus free field . Examp 8 =abcdM. The graph is then **12=1 ,=1PbdPc aWe shall also consider rational expressions over any Ia actuanocal embedding of into seen as follows: let tional ace in it the skew field D, and say that such an expression is evaluable in D if it can be evaluated without inversion of 0. f the elements of D appering in the rationl expression lly in a subring R of D, we say that the expression is over R. There is a caniare aratA ny ray 1Tser, which can be S be aies, we reploperation *T b1, which is age of ; then evaluaone obtains a rational expressible in and represents the imon in  S under the embedding . Thus, ratA  each rational S. LAVALLE´E Copyright © 2011 SciRes. APM 109language and each rational series is naturally an element of the free field. See . By Theorem 1 of , the can be evaluated in the stochastic free field. n now state the main result of this paper. rem 9 Let iP We caTheo1,=ij ij naM be a generic stochastic noncommutative matrix. Let defined as abiove. Then 11 1111,,= ,,nnP bePPPP  M (2) In other words, 111,,nPP is the left eenvector associated to the eigenvalueoigbe the m 1 f M. Example 10 (Continued.) Let be S atrix with =1ab and =1cd. We show that 11 1112 12,=,MPPPP M From this systehe two following equa- m, we deduce ttions: 11= 1Pa P11 (4) n (3), we have  (5) *12=dSince we again obtain Equation (5), it follows that Equation (4) is also satisfied. Proof of Theorem 9. Let be the matrix of the automaton and be the languastarting fr and not passing through . We have ,nwhere Let 12 1dP (3) P1221=aPdP 11 From the Equatio 11 112 11=Pa PdP   11211=1PdPa  **21=dP aP* *1=1dcaabd**  ** *** *1= 1daa ad  d da***=daad Hence, Equation (3) is satisfied. From the Equation (4), we have 11121PaPdP  11121=1Pa  **12=aP dP 1,=ij ij naMiP omge of the labels of the paths i i=1=,=1,,niijjPLi =1 =1,1, =kkkjif ij==,nnijik kjijikkjLLaaLaifi j iS be the system composed of the Equa- tions , inniP1,1,1,, , ,,iiiiiLLL L .1 Multlying to the left by ipiP each equation ofwe obtain the system n iS, 1,1:iiijjjiiTPa1111,1=,=1,,nniij iikkjkkiPPLiPLa1=iijPL Let i ijQPL, then we have n1=ij11,11,1=,=1,,:=iijjjiinijiijik kjkkiPQiTQPa Qan This system transforms into the following system We show that 11,1=,=1,,niijjjiiiijij jjPQinUPaQ aQa 1:0=n 1, ,1ikkjkkikj1111=1nkkkPa P Define 1 (6) We show that . So Equation (6) is converted into the following Equation 0 (7) ua- tion (7) and the systems . We have the matrical representation 11=kkkRPa1=1nP =0R111111=21=nkkkRP aPa Consider the system of the 21n Equations: Eqn1,,UUn1111213 122123211,1,,,,,,,,,,,,, =nnnn nnRP QQQP QQQPQ Q  Eλ , S. LAVALLE´E Copyright © 2011 SciRes. APM 110 12aa,1 ,111121, 11, 1121222, 12, 121,11,21, 11, 11,1,11,21, 11,11,,1,2, 1, 111=111mmmm mmmnmmmmmm mmmmmmnmmmmmmmnnnnm nmnna aaaaa aaaaa aaaa aaaaaa aaaaa aa nnA   Ewhere =1, ,mn and =(0, ,0,1, ,1)ntimes  . is a square mder atrix of or21n. We have 1=,RE 20,,0)Tn times and =(1,=0EF. The orde this r of matrix is . Moreover, with the stochastic identities, we have 22niBwhere the are defined by nnna 12,1 ,11121,11,11=1,12122, 12,12=1, 21,11,1,11,=1,2,11,11,21, 11,1,=1, 1,1 ,2mmmm mmmnnkmmkknkm mkknmmkmm mkkkmnmmmmmk mnkkmnnaaa aaaaa aaaaa aaaa aa aaa   ,1 ,1=1,nnm nmnkkknaa  1, 2=mmaaBaa a S. LAVALLE´E Copyright © 2011 SciRes. APM 111 To show that , we show that the matrix =0RF is hollow. For this, we apply elementary operations on the rows and on thns ofe colum FStep 2. We want a until F coa ntains st block of 0 s uch that2n3st . Step 1. We eliminate the first rowe and the last column of F. We obtain the square matrix 1F of order ). 21n (22nst nn block of 0 in the right upper corner of 1F. For this, we consider the row of j-th1F (corresnding t- row ). By structio , the first index of the elements of the ro t weer the first c. oeth the ve blocks pon ofw ofits eletion ofh hao theblock isents is We repeatrst index e j jblock jth. Now,2B. Such a this quals to of csuch that this ratioj in Bonsi row exists byn w the 1diconj-row co31Bhis which passes in the m 2Bthe fith index of nstrurow whic,, pnBBw of. We add each one of these rows to the j-th ro 1F. It follows that the atrix nar last elee all equal to we subtract the last row ments of the 1. Fromj each 1first rows of this new m jrow of the first rows, of F. We obtain the following matrix where 1,11 12, 12, 12mmnm na aaaa 112 1,=1,121 2=1, 2,1,2,, 1,1=1,=nkkknkmkkmnnn nmnmkknaa aaaaa aa C nka. We reduce=2,,mn 2F 3in removing the last line column if we start from the end. We atrix and the have the nthfollowing mF of order 2n, (1stn ). Step 3. Consider the rows of iL3F, whereand . We want thathe lastmews to be e to >in ele- nm0modinnts of these rot quals(1)t n0. Le=ik, ation wher , then ne mn01we apply the transform(1) (1)=iknknmkLLL L, to obtain the matrix =2,,mnwhere, for all , 1,=,=,=mmmijmijij ijnjijndbdmmbbDB (1)nWe remove the rows where and iL>inni and the (1n) last columns of 4F to obtain the matrix 5F of order (st 2n2n). S. LAVALLE´E Copyright © 2011 SciRes. APM 112 where mD is the matrix obtained from mD in remo- Step 4. Denote by the first column of the matrix and byn of ving the last row. 1mC the colummC md 5F, which is the ex- tension of . We to be transformed into a of. For this, we apply the 1m, C 0want , ,mn1mC vectorop=2eration 23jndcc c, where is the itich colum 3n of F. Finally we want that the first n elements t columnof th of e firs5F to transformatibe all 0; for this, we apply the on 12cc cn. We obtain the matrix 6F of order ) 21 (stnn22nn where mC is the mtrix obtaained from by remfirst column. Step 5. We want each will be transformed into a m colummCoving the mC atrix of 0. To each n q of m, denoted mCqC correspo ding to the ,*,,*Tqn column =qmdC of 6F. By construction of (hence of ), there exists a column of mCmC qb6F such that *=,*,,TqqmbC. Hence, for all, we calculate . We obtain the matrix qqqdb7F of). order 21nn (22nnst Step 6. Permuting tst and the nth column of 7he firF, we x 8obtain the matriF of der 2nor 1n (22stnn). where 1B is the 1Bblock obtained from by removing . We find a 1block of 0 upper cornthe first columnlocated in the right 211nn2)(1nn  er and 228= 21=dimnn nnF It follows that F is a hollow. From Corollary 7, Let 1is Example 11. 111=1== =0nkkkPa PRE. Hence 11,,P 11nPthe left eigenvector associated to 1.  11 1213aa21a22 2331 3233=aaa aaaM be a generic in ws nondecommutative matrix hich the variables satisfy the stochastic intitie123ii iaa a=1e associate theutomn , =1,2,3i. Wat ao Let 1P be the language recognized by automaton the Let ijL be the language of start from i to j. We have 1111213=,PL LLwords which  where 11 =1L S. LAVALLE´E Copyright © 2011 SciRes. APM 113stem each equation of 1212 1222 1332=LaLaLa 1313 12231333=LaLaLa We have the sy 12 13:=SLaLaL112 12122213321313 1223 131=PLLaLaLaLa  33Multiplying to left by 1P1S1, we obtain the system Define , then We obtain aPermuting the index of the equations of , we obtain the two following systems aaTo show that1U 12212312121123 31223211323331=:0= 10= 1PQQUPaQaQPaQaQ a 2 21113313213331311132 211332311232221=:0= 10= 1PQQUPaQaQPaQaQ a 11 111111 1211=:=PPLPLTPLP 1121131 1121122211332111 11 13113112231 1333=aPLaPLaPLPa PLaPLa   111111123 123,,= ,,PPP PPPM111, it suffices to prove that 1111221 3311=Pa Pa PaP . Define 111=ij ijQPL111111221331 1=RPa PaPa P then 1112131112 112 1222 1332113113122313331=:==PQQTQPa QaQaQPaQaQa 1111112213311=RPaPa Pa  0Under the matrical system, we have 11112 13221 23331 32,1,,,,,,,, =RPQQP QQP QQEλ  11121311112122213 321=:0= 1PQQUPaQa where Q= 0,0,0,0,0,0,0,1,1,1λ 1113122313330= 1PaQaQ a and 0000000111aWe have 11 121322 2332 332121 2311 1331 333131 3211 1221 221000000001 0000100 10000100 100001000 0001=000 10001000 100010000 0000000 10000 100aa aaaaaaaaaaaaaaaaaaE 000 1=RE,,0,0,0,0,0,0,0 where and E with stochastic iden- tities equal to 0000000111a = 1,0,0T112 1312 1321 23233231 322121 23133131 323131 3212 13122121 23000000000000100 0000100 00001000 00010 0010000 0010000 0000000 0000000 00aa aaaa aaaaaaaaa aaaaaaaa aaaa    12 1300 0 S. LAVALLE´E Copyright © 2011 SciRes. APM 114 Let =0EF, we show that F is hollow. 1000000000a 000000111a,0000001aa aaaa aaa aa11 12 13121321 23233231322121 23133131 323131 3212 13122121 2310000000000 00 00001000 000010000 0001000 00 00100 00 00100000 00100000001000000010000000111aa aaaa aaaaaaaaaaaaaaaaaaaaF0                 00010a12 13=012 13121321 23233231322121 2312 1313131 323131 3212 13122121 230000100 0000100 00001000 000100 00 001=00 0010000001000 0 000000 0 000000000011aa aaaa aaaaaaaaa aaaaaaa aaaa F 1300a15810 2491036710,,LLLL LLLL LLLL 12 13121312 131312 13122121 232321232121 23313231 323131 32313221 2312 131323131 323131 3212 1312000000000100 00 001=00 00 0010000 00000000aaaa aaaaaaaaaaaaa aaaaaaaaaaaaaa F2121 23010000000000000011aaa                  21 00a S. LAVALLE´E Copyright © 2011 SciRes. APM 1150011a , a aaa , aaa1213 12131213131213 122121232321232121 23313231 3231313231322121 2331213133131 323131 3212 131200000000 001=00000 1000 00 010000 000 0 0 0000 0 0 0aa aaaaaaa aaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa F212123 01aa                 46 56 79 89,,,LLLL LLLL 12 13121312 131312 13122121 232321232121 23313231 323131 3231322121 312331324 1213311331323131 323131 21000000000 000=00000 0000 0001000 00aa aaaaaaaaaaaa aaa aaaaaa aaaaaaaaaaaaaaaaaaaaaaa   F3221 2312 13211221232121 230000 0000000 00001aaaaaaaaaaa                  12 13121312 131312 13122121232321232121 23313231 323131 323132521 213123313212 13 3113 31323131 213221 2312 1321=000000 00000 0000 000aa aaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaa   F1221 23aa             123 423 623,,CCCCCCCCC 12 13131221 23232321 2332313231 3232621213123313212 13 311331323131 213221 2312 1321122123000000000=000 000 00000 0000 000aa aaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa               F S. LAVALLE´E Copyright © 2011 SciRes. APM 116 aaaa We have a block of zero with . It follows that 53 72,,CC CC12 1321 23233231327212131 2331321213 31 13 31 323131 213221 2312 1321122123000000000000000=000 000 00000 0000 000aaaa aaaaaaaaaaaaaaaaaaaaaaaaa              F 13CC 13 122321 2331 32328 21213123313212 13 311331323131 213221 2312 1321122123000000000000000=0 0000000 000000000 0aaaaaaa aaaaaaaaaaaaaaaaaaaaaaaa              F 35 8>7F PaThis pris hollow. By C11221Paalloorollary 7, 0. Theorem 12. Let be defined as above, then in we have (8) Proof. Define 1 11113311==PaPRoof ws us to prove the next result.iP  1=1=1niiP 1=1=1niiPR we will show that We replace the first column of the matrix 1=1=1niiPR E by the vector and the first element of by 1. Denoted1,1,0, ,0,1,0, ,0, ,1,0, ,0T  λ E and λ these two matrices. Consider the matrix =0EF. We repeat each step of the proof of Theorem 9, except for the last operation of step 4, which concerns the first column. Again, we show that F =0.is hollow, and hence, by Corollary 7, we have This proves Equation (8). Example 13. (Continued.) R11 11*121111* 1*111 111=1byEquation(5)==1=PP PPadPPbdPbd PP   =1 4. Conclusions One of the contribution of this article is the using of elementary operations on the lines and the columns. This method provides a new way to obtain some results in skew field theory with a minimum of knowledge of this theory. Moreover, Theorems 9 and 12 are proved exactly the same way. 5. Acknowledgements I am obliged to thank to Christophe Reutenauer for his comments, corrections and suggestions. 6. References  S. Lavall´ee, D. Perrin, C. Reutenauer and V. Retakh, “Codes and Noncommutative Stochastic Matrices,” To S. LAVALLE´E Copyright © 2011 SciRes. APM 117appear, 2008.  A. Broder, “Generating Random Spanning Trees. 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Malcolmson, “A Prime Matrix Ideal Yields a Skew Field,” Journal of the London Mathematical Society, l. 18, 1978, pp. 221-233.  P. M. Cohn, “Free Rings and Their Relations,” Academic Press, Salt Lake City, 1971.  P. M. Cohn, “Skew Fields: Theory of General Division Rings,” Encyclopedia of Mathematics and Its Applica-tions, Cambridge University Press, Cambridge, 1995.  M. Fliess, “Sur le Plongement de l’alg`ebre des S´eries Rationelles non Commutatives dans un Corps Gauche,” Proceedings of the National Academy of Sciences, Paris, 1970. teVo