ON THE MONOTONICITY OF THE DITTERT FUNCTION ON CLASSES OF NONNEGATIVE MATRICES

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1 Bull. Korean Math. Soc. 30 (993), No. 2, pp ON THE MONOTONICITY OF THE DITTERT FUNCTION ON CLASSES OF NONNEGATIVE MATRICES GI-SANG CHEON. Introduction Let n denote the set of all n n doubly stochastic matrices, and let J n denote the n n matrix all of whose entries are /n. The permanent of a real n n matrix A = [a ij ]isdefinedby per A = σ a σ() a nσ(n) (.) where σ runs over all permutations of {,,n}. For {,,n}, let σ (A) denote the sum of all subpermanents of order of A. The famed van der Waerden - Egoryĉev - Faliman theorem [3],[4] asserts that the permanent function attains its minimum over n uniquely at J n. In [4], Friedland and Minc remared that a stronger version of this theorem is the following. MONOTONICITY CONJECTURE. The permanent function is monotone decreasing on the line segment from A n to J n. It is often referred to as the monotonicity of permanent (abb. MP). MP has been proved for several classes of matrices in n [5],[6],[7],[9], [0],[3],[5]. A conjecture related to MP is the following one proposed by Doović[2]: Received July 25, 992. Revised March 30,993.

2 Gi-Sang Cheon DOKOVIĆ CONJECTURE. LetA n.then σ (A) (n +)2 σ (A), n = 2,,n (.2) with equality holds if and only if A = J n. The Doović conjecture for 3 was proved by Doović himself [2]. In [], Massoud Male-Shahmirzadi has revealed a connection between the monotonicity conjecture and the Doović conjecture by showing that THEOREM A. If A n satisfies the Doović inequality (.2) then MP holds for A. For a positive integer n, letk n denote the set of all real nonnegative n n matrices whose entries have sum n. For X K n with row sums r,,r n and column sums c,,c n,let ϕ(x) = r i + c j per X. (.3) Then ϕ defines a real valued function on K n. We shall call ϕ the Dittert function. The following conjecture [2] due to E. Dittert is still open. DITTERT CONJECTURE. The Dittert function attains its maximum over K n uniquely at J n. Clearly the Dittert conjecture is a generalization of van der Waerden - Egoryĉev - Faliman theorem. For {,,n},letq,n denote the set of all strictly increasing integer sequences of length chosen from,,n. For α, β Q,n,andforan n nmatrix A, leta[α β] denote the submatrix of A lying in rows α and columns β, anda(α β) is the complement of A[α β]ina. Let ϕ denote the real valued function defined on K n by ϕ (A) = r i + c j per A[α β] (.4) i α j β α,β Q,n j= 266

3 On the monotonicity of the Dittert function where r i and c j are the sum of all the entries in row i and the sum of all the entries in column j of A K n, respectively. Note that ϕ n = ϕ. We call ϕ the -th sub-dittert function. In this paper, we study the monotonicity of the Dittert function (abb. MD) on the line segment from A K n to J n generalizing both the Dittert conjecture and the Monotonicity conjecture for permanent, and obtain a sufficient condition on A K n for which the MD holds. It is also proved that if A K n satisfies the Doović inequality (.2) then MD holds for A, and a subclass of K n for which MD holds is found. 2. Some Preliminary Lemmas For {,,n},lets denote the -th elementary symmetric function of R n, i.e., S (x) = x i (2.) for x = (x,,x n ) T R n. α Q,n To study the monotonicity of the Dittert function, we need the following results concerning elementary symmetric functions and sub-dittert functions. The following lemmas are due to Cheon and Hwang []. LEMMA 2.. [] Let A K n.then i α ϕ 2 (A) ϕ 2 (J n ). (2.2) LEMMA 2.2. [] Let x = (x,,x n ) T be a nonnegative real vector such that x + +x n =nand let {,,n}.then S (x) ( ) n. (2.3) LEMMA 2.3. [4] For {,,n}, the function S /S is Schurconcave on the set of all positive real vectors. 267

4 Gi-Sang Cheon COROLLARY 2.. Let x = (x,,x n ) T be a positive real vector such that x + +x n =nand let {,,n}.then S (x) S (x) n +. (2.4) Proof. The inequality (2.4) follows directly from Lemma 2.3 and Lemma 2.2. In addition to these lemmas, we mae use of the following theorem due to Massoud Male-Shahmirzadi. LEMMA 2.4. [] Let A be an n n matrix and x a real number. Then per (A + nxj n ) = (n )!σ (A)x n. (2.5) =0 3. Monotonicity of The Dittert Function Let A K n have row sum vector R and column sum vector C. Then for each =,,n,ϕ (A)can be written as ϕ (A) = ( ) n S (R) + ( ) n S (C) σ (A). (3.) For an n n matrix A and for δ := n!n /!n n,let λ (A):=ϕ (A)+( δ )σ (A). (3.2) We are now ready to prove one of our main theorems. THEOREM 3.. Let A K n satisfy the condition λ (A) ( ) n + 2 λ (A), =,,n. (3.3) 268

5 Then MD holds for A. On the monotonicity of the Dittert function Proof. Let A be a matrix on K n with row sum vector R = (r,,r n ) T and column sum vector C = (c,,c n ) T. For real θ, leta θ =( θ)a +θj n and r i (θ) = r i + ( r i )θ, c j (θ) = c j + ( c j )θ for i, j =,,nand let R θ = (r (θ),,r n (θ)) T and C θ = (c (θ),,c n (θ)) T.Then ϕ(a θ ) = r i (θ) + c j (θ) per A θ. (3.4) j= We prove that if A K n satisfies the condition (3.3) then ϕ (A θ ) 0forthe interval 0 <θ<. Let nx := (0 <θ<).then We define and r i (θ) = θ θ A θ = +nx (A + nxj n), + nx (r i + nx) and c j (θ) = g(x ) := h(x) := r i (θ) = ( + nx) n c j (θ) = ( + nx) n j= p(x) := per A θ = + nx (c j + nx). (r i + nx), (c j + nx), j= ( + nx) n per (A + nxj n), f (x) := ϕ(a θ ) = g(x) + h(x) p(x) on the interval 0 <θ<. Then we get g (x) = n 2 ( + nx) n+ (r i + nx) + ( + nx) n (r i + nx) n r i + nx. (3.5) 269

6 Gi-Sang Cheon We compute first that and (r i + nx) (r i + nx) = r i + nx = S (R)(nx) n (3.6) =0 (n + )S (R)(nx) n. (3.7) = From (3.5), (3.6) and (3.7) it follows that g (x) = ( + nx) n+ { } (n +)S (R) S (R) n n + x n.(3.8) = Similarly we can show that h (x) = ( + nx) n+ { } (n +)S (C) S (C) n n + x n.(3.9) = On the other hand, from (2.5), we get p (x) = ( + nx) n+ (n )!{(n + ) 2 σ (A) nσ (A)}x n. = Thus from (3.8), (3.9) and (3.0), we get (3.0) where ˆT (A) f (x) = ( + nx) n+ T (A)x n (3.) = = n n + {(n + )(S (R) + S (C)) (S (R) + S (C))} + (n )! { nσ (A) (n + ) 2 σ (A) }. (3.2) 270

7 On the monotonicity of the Dittert function From (3.), we have ( ) n S (R) + S (C) = (ϕ (A) + σ (A))/. (3.3) By an elementary computation, we get, from (3.2) and (3.3) { (n ) T (A) = nn ( n λ (A) λ (A)}. (3.4) ) Hence from (3.) and (3.4), if ( ) n + 2 λ (A) λ (A) then f (x) 0, which completes the proof. In the Theorem 3., if A is restricted to be in n then the condition (3.3) coincides with the Doović inequality (.2). Note that if = n in (3.3) then where ϕ(a) n 2 ϕ ij (A) = i r + j ϕ ij (A) (3.5) j= c per A(i j). In [8], it was shown that not every matrix A K n satisfies the condition (3.5). But we guess that MD holds for every matrix in K n satisfying (3.5). For an n n matrix A, and for each =,,n,let ( ) n + 2 T (A) := λ (A) λ (A). (3.6) A simple computation shows that T (J n ) = 0 for each =,,n.the Theorem 3. just says that if T (A) 0for=,,n,then MD holds for A K n. 27

8 Gi-Sang Cheon THEOREM 3.2. The condition (3.3) holds for 2. Proof. The case = is trivial. To prove the theorem for = 2, let A be a matrix on K n with row sum vector R and column sum vector C. Then ( ) n 2 T 2 (A)=λ 2 (A) λ (A) 2 =ϕ 2 (A)+( δ 2 )σ 2 (A) ( ) n 2 ( 2 n! ). (3.7) 2 n n By an elementary computation, we get, from (3.) and (3.7) ( ) ( ( )) n n T 2 (A) = δ 2 (ϕ 2 (A) ϕ 2 (J n )) + ( δ 2 ) S 2 (R) + S 2 (C) Thus from (2.2) and (2.3), it follows that which completes the proof. T 2 (A) 0, As a corollary to Theorem 3.2, it follows that if A K n satisfies the condition (3.5) then MD holds for n 3. In the following, we find a subclass of K n for which MD holds. THEOREM 3.3. For an n n diagonal matrix D and for any n n permutation matrices P and Q, if A is a matrix of the form A = PDQ K n,thenmd holds for A. Proof. First, let A be a positive diagonal matrix on K n with row sum vector R and column sum vector C. Note S (R)=S (C)=σ (A)for each =,,n.then ( λ (A) = S (R) ( n ) δ ). (3.8)

9 On the monotonicity of the Dittert function Thus we get ( ) n + 2 T (A) = λ (A) λ (A) {( S (R) = S (R) S (R) n + )( ( ) ) n 2 δ ( } n + )( )δ n Hence from (2.4), it follows that for =,,n. T (A) 0 Now let A be a nonnegative diagonal matrix on K n with A := diag(a,,a i,0,,0), and for a sufficiently small ɛ > 0, let A(ɛ) := diag(a ɛ,,a i ɛ, ɛ,,ɛ )where ɛ = iɛ/(n i). ThenA(ɛ) is a positive diagonal matrix on K n and it follows that for each =,,n, T (A) = lim ɛ 0 T (A(ɛ)) 0. It shows the conclusion also holds for nonnegative matrices, hence the proof is completed. Theorem 3.3 has the following corollary as a special case. COROLLARY 3.. MP holds for any permutation matrix. THEOREM 3.4. If A K n satisfies the Doović inequality (.2) for each =,,n, then MD holds for A. Proof. Let A K n. What we proved in the proof of the Theorem 3.3 enables us to assume that both the row sum vector R and the column sum vector C of A are positive. Note that ( )( ) n λ (A) = S (R) + S (C) δ σ (A). (3.9) 273

10 Gi-Sang Cheon We get from (3.9) where ( ) n + 2 T (A) = λ (A) λ (A) ( )( n S (R) = S (R) S (R) n + ) ( )( n S (C) + S (C) S (C) n + ) δ D (A), From (2.4), it follows that D (A) := σ (A) (n + )2 σ (A). n T (A) δ D (A). Thus if D (A) 0 for each =,,n,then T (A) 0, which completes the proof. Note that our Theorem 3.4 is a direct generalization of the Theorem A. References. G.-S. Cheon and S.-G. Hwang, Maximization of a matrix function related to the Dittert conjecture, Linear Algebra Appl. 65(992) D.Ẑ. Doović, On a conjecture by van der Waerden, Mat. Vesni (9) 4 (967), G.P. Egoryĉev, A solution of the van der Waerden s permanent problem, Kirensi Inst. of Phisics, Acad.Sci.SSSR, Preprint IFSO-3M Kransnojars (980). 4. D.F. Faliman, A proof of van der Waerden s conjecture on the permanent of a doubly stochastic matrix, Mat.Zam. 29 (98), S. Friedland and H. Minc, Monotonicity of permanents of doubly stochastic matrices, Lin.Multilin. Alg. 6 (978),

11 On the monotonicity of the Dittert function 6. S.-G. Hwang, On the monotonicity of the permanent, Proc.Amer.Math.Soc.06 (989), , The monotonicity and the Doović conjectures on permanents of doubly stochastic matrices, Linear Algebra Appl.79 (986) S.-G. Hwang, M.G. Sohn and S.J. Kim, The Dittert s function on a set of nonnegative matrices, to appear in Internat. l J. of Math. and Math. Sci.. 9. K.-W. Lih and E.T.H. Wang, Monotonicity conjecture on permanents of doubly stochastic matrices, Proc.Amer.Math.Soc. 82 (98), D.London,Monotonicity of permanents of certain doubly stochastic matrices, Pacific J.Math. 95 (98), Massoud Male Shahmirzadi, On a conjecture by D.Ẑ. Doović, Linear Algebra Appl.30 (980), H.Minc, Theory of permanents , Lin.Multili n.alg. 2 (983), V.A.Phoung, Behavior of the permanent of a special class of doubly stochastic matrices, Lin.Multilin. Alg. 9(980), I.Schur, Uber eine Klasse von Mittelbildungen mit Anwendungen die Determinanten Theorie sitzungsber, Berlin Math.Gesellschaft, 22(923), R.Sinhorn, Concerning the question of monotonicity of the permanent on doubly stochastic matrices, Lin.Multilin.Alg. 8 (980). DEPARTMENT OF MATHEMATICS,SUNG KYUN KWAN UNIVERSITY, SUWON , KOREA 275