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Generalized Power Symmetric Stochastic Matrices 1989 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<mare positive integers. 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,wherep<mare positive integers. We give a characterization of generalized power symmetric stochastic ma- trices, thereby generalizing Sinkhorn's result. The proof makes nontrivial use of n the machinery of generalized inverses. In view of the fact that (A )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 (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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1988 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(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<m are positive integers. Then the× following assertions are true: (i) rank(Ap)=rank(Ak),k p, (ii) index(A) p, ≥ (iii) index(Ap)=1≤ , (iv) Ap is an EP matrix, p # p (v) (A ) =(A )†. Proof. (i) Clearly, rank(Ap) rank(Ak) rank(Am)forp k m.Since (Ap)T=Am,wehave ≥ ≥ ≤ ≤ rank(Ap)=rank(Ap)T = rank(Am) and thus rank(Ap)=rank(Ak),p k m. It follows that rank(Ap)= rank(Ak),k p. ≤ ≤ (ii) By (i), rank(A≥p)=rank(Ap+1) and hence index(A) p. (iii) By (i), rank(Ap)=rank(A2p) and hence index(Ap)=1.≤ (iv) Since rank(Ap)=rank(Am), the column space of Ap is the same as that of Am.Also,since(Ap)T =Am, the column space of Am is the same as the row space of Ap, written as a set of column vectors. Therefore, the column spaces of Ap and (Ap)T are identical and Ap is an EP matrix. (v) It is known (see, for example, [3], p. 129) 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 (Ap)T = Am,wherep<mare positive integers. Let γ = m× p.ThenAp=ApAiγ (Aiγ )T for any integer i 0. − ≥ Proof. We have (1) Ap =(Ap+γ)T =(Ap)T(Aγ)T =Ap+γ(Aγ)T =AγAp(Aγ)T =Aγ(AγAp(Aγ)T)(Aγ )T = A2γ Ap(A2γ )T . Repeating the argument, we get Ap = Aiγ Ap(Aiγ )T = ApAiγ (Aiγ )T for any i 0 and the proof is complete. ≥ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GENERALIZED POWER SYMMETRIC STOCHASTIC MATRICES 1989 Lemma 3. Let A be a nonnegative n n matrix and suppose (Ap)T = Am,where p<mare positive integers. Then (Ap×)# exists and is nonnegative. Proof. By Lemma 1 (iii), index(Ap) = 1 and therefore (Ap)# exists. Let γ = m p. Setting i = p in Lemma 2, we get − Ap = ApApγ (Apγ )T = ApApγ A(p+γ)γ in view of (Ap)T = Am. Thus, since γ 1, ≥ p 2p p(γ 1) (p+γ)γ A = A A − A 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 p<mare positive integers. Then there× exists a permutation 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 2 23 ··· 3 II. · where there exists a positive integer u such 6 7 6 · 7 6 0 Cd 1,d 7 6 ··· − 7 6 Cd1 0 0 7 6 ··· u T7 u T 4that (C12 C23 Cd1) =x1x15, ,(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 2 ··· 3 · 6 7 6 · T 7 6 00 xdx 7 6 ··· d 7 4 5 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1990 R. B. BAPAT, S. K. JAIN, AND K. MANJUNATHA PRASAD ^ J(n , ,n ) will denote the n n matrix 1 ··· d × 1 0 Jn n 0 0 n2 1 2 × 1 ··· 00Jnn 0 2 n32×3··· 3 · 6 7 6 · 1 7 6 00 Jn n 7 nd d 1 d 6 1 ··· − × 7 6 Jn n 0 0 7 6 n1 d× 1 ··· 7 4 5 where the zero blocks along the diagonal are square, of order n1,n2, ,nd, respectively. We remark that if d =1,thenJ^ = 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 p<mbe 1 ··· d positive integers. Let p = µ mod d, m = µ0 mod d where 0 µ d 1, 0 µ0 ^ ≤ ≤ − ≤ ≤ d 1.LetS=J(n, ,n ). Then the following conditions are equivalent: − 1 ··· d (i) (Sp)T = Sm; (ii) (a) d divides both p and m,or(b)ddivides neither p nor m, µ + µ0 = d and n = n , 1 i d; i i+µ0 ≤ ≤ (iii) (a) d divides both p and m,or(b)ddivides neither p nor m, µ + µ0 = d and n = n , 1 i d,whereδ=(µ, µ ) is the g.c.d.
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