PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 7, Pages 1987–1994 S 0002-9939(99)05126-6 Article electronically published on March 17, 1999

GENERALIZED POWER SYMMETRIC STOCHASTIC MATRICES

R. B. BAPAT, S. K. JAIN, AND K. MANJUNATHA PRASAD

(Communicated by Jeffry N. Kahn)

Abstract. We characterize stochastic matrices A which satisfy the equation (Ap)T = Am where p

1. Introduction An n n matrix is called stochastic if every entry is nonnegative and each row sum is one.× The transpose of the matrix A will be denoted by AT .AmatrixAis doubly stochastic if both A and AT are stochastic. Sinkhorn [5] characterized stochastic matrices A which satisfy the condition AT = Ap,wherep>1 is a positive integer. Such matrices were called power symmetric in [5]. In this paper we consider a generalization. Call a square ma- trix A generalized power symmetric if (Ap)T = Am,wherepprobability of an event to change from the state i to the state j in n units of time, the reader may find it interesting to interpret the condition (Ap)T = Am on the matrix A =(Ai,j ). The paper is organized as follows. In the next section, we introduce some defini- tions and prove several preliminary results. The main results are proved in Section 3.

2. Preliminary results If A is an m n matrix, then an n m matrix G is called a generalized inverse of A if AGA =×A.IfAis a square matrix,× then G is the group inverse of A if AGA = A, GAG = G and AG = GA. We refer to ([1], [2], [3]) for the background concerning generalized inverses. It is well known that A admits group inverse if and only if rank(A)=rank(A2), in which case the group inverse, denoted by A#, is unique.

Received by the editors October 22, 1997. 1991 Mathematics Subject Classification. Primary 05B20, 15A09, 15A51. This work was done while the second author was visiting Indian Statistical Institute in November-December 1996 and July-August 1997. He would like to express his thanks for the warm hospitality he enjoyed during his visits as always. Partially supported by Research Chal- lenge Fund, Ohio University.

c 1999 American Mathematical Society

1987

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If A is an n n matrix, then the index of A, denoted by index(A), is the least positive integer×k such that rank(Ak)=rank(Ak+1). Thus A has group inverse if and only if index(A)=1. If A is an m n real matrix, then the n m matrix G is called the Moore-Penrose inverse of A if× it satisfies AGA = A, GAG× = G, (AG)T = AG, (GA)T = GA.The Moore-Penrose inverse of A, which always exists and is unique, is denoted by A†. A real matrix A is said to be an EP matrix if the column spaces of A and AT are identical. We refer to [1] or [3] for elementary properties of EP matrices. Lemma 1. Let A be a real n n matrix and suppose (Ap)T = Am,wherep

Lemma 2. Let A be an n n matrix such that (Ap)T = Am,wherep

Repeating the argument, we get Ap = Aiγ Ap(Aiγ )T = ApAiγ (Aiγ )T for any i 0 and the proof is complete. ≥

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Lemma 3. Let A be a nonnegative n n matrix and suppose (Ap)T = Am,where p

o p(γ 1) (p+γ)γ where A is taken to be the identity matrix. Let B = A − A .Thenit follows from the previous equation that ApBAp = Ap. Also, since B is a power of A, ApB = BAp.Thus(Ap)#=BApB is also a power of A and hence is nonnegative.

Lemma 4. Let A be a stochastic n n matrix and suppose (Ap)T = Am,where ppermutation matrix P such that PAPT is a direct sum of matrices of the following two types I and II: v T I. C11 where C11 = xx ,x>0 for some positive integer v. 0 C12 0 0 00C··· 0  23 ···  II. · where there exists a positive integer u such    ·   0 Cd 1,d   ··· −   Cd1 0 0   ··· u T u T that (C12 C23 Cd1) =x1x1, ,(Cd1 C12 Cd 1,d) = xdxd for some positive vectors··· x , ,x . ··· ··· − 1 ··· d p p # p # p Proof. Let C = A (A ) . Since (A ) =(A )† by Lemma 1, C is symmetric. By Lemma 3, (Ap)# is nonnegative and hence C is nonnegative. Also, C is clearly idempotent. As observed in the proof of Lemma 3, (Ap)# is a power of A and hence C is also a power of A. The nonnegative roots of symmetric, nonnegative, idempotent matrices have been characterized in [4], Theorem 2. Applying the result to C, we get the form of A asserted in the present result. We remark that according to Theorem 2 of [4], a type III summand is also possible along with the two types mentioned above. However, a matrix of this type is necessarily nilpotent and since A is stochastic, such a summand is not possible.

Remark. Using the notation of Lemma 4, it can be verified that the type II sum- mand given there has the property that its (du)th power is

T x1x1 0 0 0 x xT ··· 0  2 2 ···  ·    · T   00 xdx   ··· d    This observation will be used in the sequel. We now introduce some notation. Denote by Jm n the m n matrix with each entry equal to one. If n , ,n are positive integers× adding× up to n,then 1 ··· d

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ˆ J(n , ,n ) will denote the n n matrix 1 ··· d × 1 0 Jn n 0 0 n2 1 2 × 1 ··· 00Jnn 0  n32×3···  ·    · 1   00 Jn n  nd d 1 d  1 ··· − ×   Jn n 0 0   n1 d× 1 ···    where the zero blocks along the diagonal are square, of order n1,n2, ,nd, respectively. We remark that if d =1,thenJˆ = 1 J . ··· (n1) n1 n1 n1 From now onwards, we make the convention that when× we deal with integers n , ,n , the subscripts of n should be interpreted modulo d. Thus, for example, 1 ··· d nd+1 = n1,n 2 =nd 2 and so on. − − Lemma 5. Let n , ,n be positive integers summing to n and let p0. Note (i) implies d divides p if and only if d divides m. We first prove (i) (ii): m ⇔ Assume d divides neither p nor m.Let(S)idenote the (i, j)-block in the partitioning of Sµ, in conformity with the partitioning of S, 1 i, j d. Similarly µ0 µ0 ≤ ≤ (S )ij will denote the (i, j)-block in S . A straightforward multiplication involving partitioned matrices shows that 1 Jn n , if j =(µ+i)mod d, (Sµ) = nµ+i i× µ+i ij 0, otherwise.  Similarly, 1 J , if j =(µ +i)mod d. µ0 n ni nµ +i 0 (S )ij = µ0+i × 0 ( 0, otherwise. Now 1 Jn n , if i =(µ+j)mod d, (Sµ)T = nµ+j µ+j × j ij 0, otherwise.  or equivalently, 1 Jn n , if j =(i µ)mod d, (Sµ)T = nµ+j µ+j × j − ij 0, otherwise. 

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µ T µ0 Thus (S ) = S holds if and only if µ0 + i =(i µ)mod d and ni = nµ0+i.Since p T m µ T µ − (S) =S holds if and only if (S ) = S 0, it follows that under the condition d does not divide p and d does not divide m, (i) holds if and only if µ + µ = d and 0 ni = ni+µ , 1 i d. Furthermore,0 ≤ because≤ d p and d m trivially imply (Sp)T = Sm, the proof of (i) (ii) is completed. | | ⇔ If (ii) holds, then ni = nµ +i = nd µ+i, 1 i d. Hence 0 − ≤ ≤ ni = nαµ βµ+βd+i = nαµ βµ+i , 0− 0− where α, β are positive integers, 1 i d. Choosing α, β such that δ = αµ0 βµ, we get n = n , 1 i d. This≤ proves≤ (iii). − i δ+i ≤ ≤ Conversely, if (iii) holds, then ni = nµ0+i since δ divides µ0, establishing (ii). This completes the proof.

3. The main results We are now ready to give a characterization of generalized power symmetric stochastic matrices. In the next result we consider matrices of index one. Theorem 1. Let A be a stochastic n n matrix with index(A)=1and let p0 for some positive v integer v.Sinceindex(C11) = 1, it follows that rank(C11)=rank(C11)=1. T p T m Suppose C11 =˜xy˜ for somex> ˜ 0,y>˜ 0. Now using (C11) = C11,itcanbe shown that C11 is symmetric and hence doubly stochastic. It is well known that a 1 positive, rank one doubly stochastic matrix must be k Jk k for some k. Now suppose S is a summand of type II and let ×

0 C12 0 0 00C··· 0  23 ···  S = ·    ·   00 Cd 1,d   ··· −   Cd1 0 0   ···    where the zero blocks on the diagonal are square of order n1, ,nd, respectively. As observed in the remark following Lemma 4, there exists a positive··· integer u such that Sdu is a direct sum of d positive, symmetric, idempotent, rank one matrices. We first claim that rank(C12)=rank(C23)= =rank(Cd1)=1.Foroth- erwise, rank(S) >d. However, rank(Sdu)=dand··· since index(S)=1, rank(S)= du rank(S ), giving a contradiction. Therefore, the claim is proved. We let C12 =

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x yT ,C = x yT , ,C = x yT,wherex,y are positive vectors. Since 1 2 23 2 3 ··· d1 d 1 i i C12,C23, ,Cd1 have rank one, we may choose xi = Jni 1, 1 i d. In view··· of the description of Sdu given earlier, we may write× ≤ ≤ (C C C )u =xxT 12 23 ··· d1 for some positive vector x.Thus (xyT xyT x yT)u =xxT 1 2 2 3 ··· d 1 T T and hence λx1y1 = xx for some λ>0. It follows that y1 = γ1Jn1 1 for some × γ1 > 0. Similarly we conclude that yi = γi Jni 1,γi > 0, 1 i d.Now, interpreting subscripts modulo d as usual, we have× ≤ ≤ T (2) Ci,i+1 = xiyi+1 T = Jn 1 (γi+1 Jn 1) i× i+1× = γi+1 Jn n i× i+1 1 = Jni ni+1 , ni+1 ×

since Ci,i+1 has row sums one, 1 i d. The remaining assertions in (ii) follow from Lemma 5. ≤ ≤ Conversely, it is clear that if S is a summand of type I or II, then (Sp)T = Sm. This completes the proof.

Corollary 1. Let A be an n n stochastic matrix of index one and suppose (Ap)T = Am where p1. Then× there exists a permutation matrix P such that PAPT is a direct sum of matrices of the following types: 1 (1) k Jk k for some positive integer k; ˆ × (2) J(`, ,`) for some positive integer `,where`occurs d times and d divides m +1···. Proof. By Lemma 1, index(A) = 1. Since the g.c.d. of 1,m is 1, by Corollary 1 there exists a permutation matrix P such that PAPT is a direct sum of matrices of types given in Corollary 1. The rest of the proof follows easily.

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Our next objective is to describe a stochastic matrix A, not necessarily of index one, which satisfies (Ap)T = Am for positive integers p

and then p p # p p # (4) A (A ) A = A (A ) (CA + NA) p p # = CA + A (A ) NA p # p = CA +(A ) CANA = CA.

Since (Ap)# is a power of A, as observed in the proof of Lemma 3, it follows that CA is also a power of A.ThusCAis nonnegative and stochastic. Theorem 2. Let A be an n n stochastic matrix. Then (Ap)T = Am,wherep

p p m m Proof. ‘Only if’ part: By Lemma 1, index(A) p and hence A = CA,A =CA . p T m ≤ Thus (CA) = CA .Sinceindex(CA) = 1 and by Lemma 6 CA is stochastic, T Theorem 1 yields that there exists a permutation matrix P such that PCAP is a direct sum of matrices of types I, II as given in Theorem 1. Let T T T PAP =[Aij ],PCAP =[(CA)ij ],PNAP =[(NA)ij ]

be compatible partitions. Note that (CA)ij =0ifi=j.SinceAij =(CA)ij + (N ) ,wegetthat(N ) 0fori=j. However,6 since C N =0,wehave A ij A ij ≥ 6 A A (CA)ii(NA)ij =0,i = j. It follows from the description of (CA)ii, in Theorem 6 s 1 and the remark following Lemma 4 that for some positive integer s, (CA)ii has

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positive diagonal entries. This, along with (C ) (N ) =0,and(N ) 0for A ii A ij A ij ≥ i=j, yields (NA)ij =0.ThusAij =0,i = j. Setting Cii =(CA)ii,Nii =(NA)ii for6 all i, the result follows. The ‘if part’ is6 straightforward. We conclude with an example. Let

1 1 002 2 001 1 A =  2 2 1 2 00  3 3   2 1   00  3 3  Then A is stochastic and (A2)T = A4. The core-nilpotent decomposition is given by A = CA + NA,where 1 1 002 2 0000 1 1 002 2 0000 CA = ,NA= 1 1 00 1 1 00 2 2  −6 6  1 1  1 1   00  − 00 2 2  6 6  Thus CA is stochastic and is of type II as described in Theorem 2. This example also shows that NA need not be nonnegative. Acknowledgement The authors would like to thank the referee for helpful suggestions and comments. References

[1] A. Ben-Israel and T.N.E. Greville, Generalized Matrix Inverses: Theory and Applications, Wiley, NY, 1973. MR 53:469 [2] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,2nd edition, SIAM, 1995. MR 95e:15013 [3] S. L. Campbell and C.D. Meyer, Jr., Generalized Inverses of Linear Transformations,Pit- man, 1979. MR 80d:15003 [4] S. K. Jain, V.K. Goel and Edward K. Kwak, Nonnegative mth roots of nonnegative O- symmetric idempotent matrices, Linear Algebra Appl. 23:37-51 (1979). MR 80m:15002 [5] Richard Sinkhorn, Power symmetric stochastic matrices, Linear Algebra Appl. 40:225-228 (1981). MR 82j:15017

Department of Mathematics, Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi 110016, India E-mail address: [email protected] Department of Mathematics, Ohio University, Athens, Ohio 45701 E-mail address: [email protected] Department of Mathematics, Manipal Institute of Technology, Gangtok, Sikkim, 737102 India

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