GENERALIZED POWER SYMMETRIC STOCHASTIC MATRICES

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 27, Number 7, Pages S (99) Article electronically published on March 7, 999 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 (A p ) T = A m where p<mare positive integers.. 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 A T.AmatrixAis doubly stochastic if both A and A T are stochastic. Sinkhorn [5] characterized stochastic matrices A which satisfy the condition A T = A p,wherep>isapositive integer. Such matrices were called power symmetric in [5]. In this paper we consider a generalization. Call a square matrix A generalized power symmetric if (A p ) T = A m,wherep<mare positive integers. We give a characterization of generalized power symmetric stochastic matrices, thereby generalizing Sinkhorn s result. The proof makes nontrivial use of the machinery of generalized inverses. In view of the fact that (A n ) i,j represents the probability 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 (A p ) T = A m on the matrix A =(A i,j ). The paper is organized as follows. In the next section, we introduce some definitions and prove several preliminary results. The main results are proved in Section 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. If A is a square matrix, then G is the group inverse of A if AGA = A, GAG = G and AG = GA. We refer to ([], [2], [3]) for the background concerning generalized inverses. It is well known that A admits group inverse if and only if rank(a) =rank(a 2 ), in which case the group inverse, denoted by A #, is unique. Received by the editors October 22, Mathematics Subject Classification. Primary 05B20, 5A09, 5A5. This work was done while the second author was visiting Indian Statistical Institute in November-December 996 and July-August 997. He would like to express his thanks for the warm hospitality he enjoyed during his visits as always. Partially supported by Research Challenge Fund, Ohio University. 987 c 999 American Mathematical Society

2 988 R. B. BAPAT, S. K. JAIN, AND K. MANJUNATHA PRASAD 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(a k )=rank(a k+ ). Thus A has group inverse if and only if index(a) =. 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 A T are identical. We refer to [] or [3] for elementary properties of EP matrices. Lemma. Let A be a real n n matrix and suppose (A p ) T = A m,wherep<m are positive integers. Then the following assertions are true: (i) rank(a p )=rank(a k ), k p, (ii) index(a) p, (iii) index(a p )=, (iv) A p is an EP matrix, (v) (A p ) # =(A p ). Proof. (i) Clearly, rank(a p ) rank(a k ) rank(a m )forp k m. Since (A p ) T = A m,wehave rank(a p ) = rank(a p ) T = rank(a m ) and thus rank(a p )=rank(a k ), p k m. It follows that rank(a p )= rank(a k ), k p. (ii) By (i), rank(a p )=rank(a p+ ) and hence index(a) p. (iii) By (i), rank(a p )=rank(a 2p ) and hence index(a p )=. (iv) Since rank(a p )=rank(a m ), the column space of A p is the same as that of A m.also,since(a p ) T =A m, the column space of A m is the same as the row space of A p, written as a set of column vectors. Therefore, the column spaces of A p and (A p ) T are identical and A p is an EP matrix. (v) It is known (see, for example, [3], p. 29) that for an EP matrix the group inverse exists and coincides with the Moore-Penrose inverse. Lemma 2. Let A be an n n matrix such that (A p ) T = A m,wherep<mare positive integers. Let γ = m p. ThenA p =A p A iγ (A iγ ) T for any integer i 0. Proof. We have () A p = (A p+γ ) T = (A p ) T (A γ ) T = A p+γ (A γ ) T = A γ A p (A γ ) T = A γ (A γ A p (A γ ) T )(A γ ) T = A 2γ A p (A 2γ ) T. Repeating the argument, we get A p = A iγ A p (A iγ ) T = A p A iγ (A iγ ) T for any i 0 and the proof is complete.

3 GENERALIZED POWER SYMMETRIC STOCHASTIC MATRICES 989 Lemma 3. Let A be a nonnegative n n matrix and suppose (A p ) T = A m,where p<mare positive integers. Then (A p ) # exists and is nonnegative. Proof. By Lemma (iii), index(a p ) = and therefore (A p ) # exists. Let γ = m p. Setting i = p in Lemma 2, we get A p = A p A pγ (A pγ ) T = A p A pγ A (p+γ)γ in view of (A p ) T = A m. Thus, since γ, A p = A 2p A p(γ ) A (p+γ)γ where A o is taken to be the identity matrix. Let B = A p(γ ) A (p+γ)γ. Then it follows from the previous equation that A p BA p = A p. Also, since B is a power of A, A p B = BA p.thus(a p ) # =BA p B is also a power of A and hence is nonnegative. Lemma 4. Let A be a stochastic n n matrix and suppose (A p ) T = A m,where p<mare positive integers. Then there exists a permutation matrix P such that PAP T is a direct sum of matrices of the following two types I and II: I. C where C v = xxt,x>0 for some positive integer v. 0 C C 23 0 II. where there exists a positive integer u such 0 C d,d C d 0 0 that (C 2 C 23 C d ) u =x x T,,(C d C 2 C d,d ) u = x d x T d for some positive vectors x,,x d. Proof. Let C = A p (A p ) #. Since (A p ) # =(A p ) by Lemma, C is symmetric. By Lemma 3, (A p ) # is nonnegative and hence C is nonnegative. Also, C is clearly idempotent. As observed in the proof of Lemma 3, (A p ) # 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 summand given there has the property that its (du)th power is x x T x 2 x T 2 0 x d x T d This observation will be used in the sequel. We now introduce some notation. Denote by J m n the m n matrix with each entry equal to one. If n,,n d are positive integers adding up to n, then

4 990 R. B. BAPAT, S. K. JAIN, AND K. MANJUNATHA PRASAD Ĵ (n,,n d ) will denote the n n matrix 0 n 2 J n n n 3 J n2 n 3 0 n d J nd n d n J nd n 0 0 where the zero blocks along the diagonal are square, of order n,n 2,,n d, respectively. We remark that if d =,thenĵ(n) = n J n n. From now onwards, we make the convention that when we deal with integers n,,n d, the subscripts of n should be interpreted modulo d. Thus, for example, n d+ = n,n 2 =n d 2 and so on. Lemma 5. Let n,,n d be positive integers summing to n and let p<mbe positive integers. Let p = µ mod d, m = µ mod d where 0 µ d, 0 µ d. LetS=Ĵ(n,,n d ). Then the following conditions are equivalent: (i) (S p ) T = S m ; (ii) (a) d divides both p and m, or(b)ddivides neither p nor m, µ + µ = d and n i = n i+µ, i d; (iii) (a) d divides both p and m, or(b)ddivides neither p nor m, µ + µ = d and n i = n i+δ, i d, whereδ=(µ, µ ) is the g.c.d. of µ and µ. Proof. We first observe that n J n n S d n 2 J n2 n 2 0 = n d J nd n d S d S=S and so S d S i = S i, for all i>0. Note (i) implies d divides p if and only if d divides m. We first prove (i) (ii): Assume d divides neither p nor m. Let (S m ) i denote the (i, j)-block in the partitioning of S µ, in conformity with the partitioning of S, i, j d. Similarly (S µ ) ij will denote the (i, j)-block in S µ. A straightforward multiplication involving partitioned matrices shows that { (S µ ) ij = n µ+i J ni n µ+i, if j =(µ+i)mod d, 0, otherwise. Similarly, Now (S µ ) ij = { n µ +i J ni n µ +i, if j =(µ +i)mod d. 0, otherwise. { (S µ ) T ij = n µ+j J nµ+j n j, if i =(µ+j)mod d, 0, otherwise. or equivalently, { (S µ ) T ij = n µ+j J nµ+j n j, if j =(i µ)mod d, 0, otherwise.

5 GENERALIZED POWER SYMMETRIC STOCHASTIC MATRICES 99 Thus (S µ ) T = S µ holds if and only if µ + i =(i µ)mod d and n i = n µ +i. Since (S p ) T =S m holds if and only if (S µ ) T = S µ, 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 n i = n i+µ, i d. Furthermore, because d p and d m trivially imply (S p ) T = S m, the proof of (i) (ii) is completed. If (ii) holds, then n i = n µ +i = n d µ+i, i d. Hence n i = n αµ βµ+βd+i = n αµ βµ+i, where α, β are positive integers, i d. Choosing α, β such that δ = αµ βµ, we get n i = n δ+i, i d. This proves (iii). Conversely, if (iii) holds, then n i = n µ +i since δ divides µ, 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. Let A be a stochastic n n matrix with index(a) =and let p<m be positive integers. Then (A p ) T = A m if and only if there exists a permutation matrix P such that PAP T is a direct sum of matrices of the following two types I and II: I. k J k k for some positive integer k; II. d d block partitioned matrices of the form Ĵ(n,,n d ) satisfying (a) d p and d m, or (b) ddivides neither p nor m such that if p = µ(mod d),m = µ (mod d) where 0 µ, µ d and δ =(µ, µ ), thenµ+µ =d, n i = n i+δ. Proof. First suppose that (A p ) T = A m. By Lemma 4 we conclude that there exists a permutation matrix P such that PAP T is a direct sum of matrices of types I, II given in Lemma 4. Let S = C be a summand of type I so that C v = xxt,x>0 for some positive integer v. Since index(c ) =, it follows that rank(c ) = rank(c v ) =. Suppose C = xỹ T for some x >0,ỹ>0. Now using (C p )T = C, m itcanbe shown that C is symmetric and hence doubly stochastic. It is well known that a positive, rank one doubly stochastic matrix must be k J k k for some k. Now suppose S is a summand of type II and let S = 0 C C 23 0 C d,d C d 0 0 where the zero blocks on the diagonal are square of order n,,n d, respectively. As observed in the remark following Lemma 4, there exists a positive integer u such that S du is a direct sum of d positive, symmetric, idempotent, rank one matrices. We first claim that rank(c 2 )=rank(c 23 )= =rank(c d )=. Forotherwise, rank(s) >d. However, rank(s du )=dand since index(s) =, rank(s) = rank(s du ), giving a contradiction. Therefore, the claim is proved. We let C 2 =

6 992 R. B. BAPAT, S. K. JAIN, AND K. MANJUNATHA PRASAD x y2 T,C 23 = x 2 y3 T,,C d = x d y T, where x i,y i are positive vectors. Since C 2,C 23,,C d have rank one, we may choose x i = J ni, i d. In view of the description of S du given earlier, we may write (C 2 C 23 C d ) u =xx T for some positive vector x. Thus (x y2 T x 2y3 T x d y T )u =xx T and hence λx y T = xxt for some λ>0. It follows that y = γ J n for some γ > 0. Similarly we conclude that y i = γ i J ni,γ i > 0, i d. Now, interpreting subscripts modulo d as usual, we have (2) C i,i+ = x i yi+ T = J ni (γ i+ J ni+ ) T = γ i+ J ni n i+ = J ni n n i+, i+ since C i,i+ has row sums one, 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 (S p ) T = S m. This completes the proof. Corollary. Let A be an n n stochastic matrix of index one and suppose (A p ) T = A m where p<mare positive integers with (p, m) =. Then there exists a permutation matrix P such that PAP T is a direct sum of matrices of the following types I and II: I. k J k k for some positive integer k; II. Ĵ (l,,l) for some positive integer l, occurring d times and d divides p + m. Proof. By Theorem there exists a permutation matrix P such that PAP T is a direct sum of matrices of types I, II given in Theorem. Consider a summand of type II, say Ĵ(n,,n d ). Let µ = p mod d, µ = mmodd. Then by Theorem, µ + µ = d and n i = n i+δ,whereδis the g.c.d. of µ, µ. Since (p, m) =and µ+µ =d, it follows that δ =. Thusn =n 2 = =n d =l,say. Thusthe summand equals Ĵ(l,,l). Also,µ+µ =dimplies that p + m =0mod d and thus d divides p + m. We now derive the main result in [5]. Corollary 2. Let A be an n n stochastic matrix and suppose A T = A m for some positive integer m>. Then there exists a permutation matrix P such that PAP T is a direct sum of matrices of the following types: () k J k k for some positive integer k; (2) Ĵ(l,,l) for some positive integer l, whereloccurs d times and d divides m +. Proof. By Lemma, index(a) =. Since the g.c.d. of,m is, by Corollary there exists a permutation matrix P such that PAP T is a direct sum of matrices of types given in Corollary. The rest of the proof follows easily.

7 GENERALIZED POWER SYMMETRIC STOCHASTIC MATRICES 993 Our next objective is to describe a stochastic matrix A, not necessarily of index one, which satisfies (A p ) T = A m for positive integers p<m. If A is an n n matrix, then recall that A can be expressed as A = C A +N A where C A, the core part of A, is of index one, N A is nilpotent and C A N A = N A C A =0. This is referred to as the core-nilpotent decomposition of A. We refer to [2] for basic properties of this decomposition. Lemma 6. Let A be a stochastic n n matrix such that (A p ) T = A m,wherep<m are positive integers. Let A = C A + N A be the core-nilpotent decomposition. Then C A is nonnegative and stochastic. Proof. By Lemma, index(a) p. ThenA p =C p A.Sinceindex(C A)=,wecan write C A = C p AX for some matrix X. Now (3) A p (A p ) # C A = A p (A p ) # C p A X = A p (A p ) # A p X = A p X = C p A X = C A and then (4) A p (A p ) # A = A p (A p ) # (C A + N A ) = C A + A p (A p ) # N A = C A +(A p ) # C p A N A = C A. Since (A p ) # is a power of A, as observed in the proof of Lemma 3, it follows that C A is also a power of A. ThusC A is nonnegative and stochastic. Theorem 2. Let A be an n n stochastic matrix. Then (A p ) T = A m,wherep<m are positive integers if and only if there exists a permutation matrix P such that PAP T is a direct sum of matrices C ii + N ii, i k, where () C ii are stochastic matrices of index,n ii are nilpotent matrices of index p with sum of entries of each row as zero, C ii N ii =0=N ii C ii, i k, and (2) Each C ii is a direct sum of matrices of types (I) and (II) as described in Theorem. Proof. Only if part: By Lemma, index(a) p and hence A p = C p A,Am =CA m. Thus (C p A )T = CA m. Since index(c A) = and by Lemma 6 C A is stochastic, Theorem yields that there exists a permutation matrix P such that PC A P T is a direct sum of matrices of types I, II as given in Theorem. Let PAP T =[A ij ],PC A P T =[(C A ) ij ], PN A P T =[(N A ) ij ] be compatible partitions. Note that (C A ) ij =0ifi j. Since A ij =(C A ) ij + (N A ) ij,wegetthat(n A ) ij 0fori j. However, since C A N A =0,wehave (C A ) ii (N A ) ij =0,i j. It follows from the description of (C A ) ii, in Theorem and the remark following Lemma 4 that for some positive integer s, (C A ) s ii has

8 994 R. B. BAPAT, S. K. JAIN, AND K. MANJUNATHA PRASAD positive diagonal entries. This, along with (C A ) ii (N A ) ij =0,and(N A ) ij 0for i j, yields (N A ) ij =0. ThusA ij =0,i j. Setting C ii =(C A ) ii,n ii =(N A ) ii for all i, the result follows. The if part is straightforward. We conclude with an example. Let A = Then A is stochastic and (A 2 ) T = A 4. The core-nilpotent decomposition is given by A = C A + N A,where 2 2 C A = , N A = Thus C A is stochastic and is of type II as described in Theorem 2. This example also shows that N A need not be nonnegative. Acknowledgement The authors would like to thank the referee for helpful suggestions and comments. References [] A. Ben-Israel and T.N.E. Greville, Generalized Matrix Inverses: Theory and Applications, Wiley, NY, 973. MR 53:469 [2] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, 2nd edition, SIAM, 995. MR 95e:503 [3] S. L. Campbell and C.D. Meyer, Jr., Generalized Inverses of Linear Transformations, Pitman, 979. MR 80d:5003 [4] S. K. Jain, V.K. Goel and Edward K. Kwak, Nonnegative mth roots of nonnegative O- symmetric idempotent matrices, Linear Algebra Appl. 23:37-5 (979). MR 80m:5002 [5] Richard Sinkhorn, Power symmetric stochastic matrices, Linear Algebra Appl. 40: (98). MR 82j:507 Department of Mathematics, Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi 006, India address: rbb@isidl.isid.ac.in Department of Mathematics, Ohio University, Athens, Ohio address: jain@math.ohiou.edu Department of Mathematics, Manipal Institute of Technology, Gangtok, Sikkim, India