Positivity properties of some special matrices Priyanka Grover ,∗ Veer Singh Panwar ,† A. Satyanarayana Reddy‡ Department of Mathematics Shiv Nadar University, Dadri U.P. 201314, India. Abstract 1 It is shown that for positive real numbers 0 < λ1 < ··· < λn, , where β(·, ·) denotes the beta function, is h β(λi,λj ) i 1 infinitely divisible and totally positive. For h β(i,j) i, the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let w(n) be the nth Bell number. It is proved that [w(i + j)] is a totally positive matrix but is infinitely divisible only upto order 4. It is also shown that the symmetrized Stirling matrices are totally positive. AMS classification: 15B48, 42A82,15B36 Keywords : Bell numbers, infinitely divisible matrices, positive semidefinite matrices, Schur product, Stirling num- bers, totally positive matrices, the beta function. 1 Introduction Let Mn(C) be the set of all n n complex matrices. A matrix A Mn(C) is said to be positive × n ∈ n semidefinite if x∗Ax 0 for all x C and positive definite if x∗Ax > 0 for all x C , x = 0. Let ≥ ∈ ∈ 6 A = [aij ] and B = [bij ]. In this paper, 1 i, j n, unless otherwise specified. The Hadamard product or ≤ ≤ the Schur product of two matrices A and B is denoted by A B, where A B = [aij bij ]. For a nonnegative r r ◦ ◦ real number r, A◦ = [aij ]. r Let A = [aij ] be such that aij 0. The matrix A is called infinitely divisible if A◦ is positive semidefinite for every real number r≥ > 0. For examples and properties of infinitely divisible matrices, see [3, 7, 15]. The matrix A is called totally positive or totally nonnegative if all its minors are positive or nonnegative respectively. For more results on these, see [12]. The main objective of this paper is to explore the above mentioned properties for a few matrices which are constructed from interesting arXiv:2002.08703v4 [math.FA] 3 May 2020 functions. Many such matrices have been studied in [3, 7, 5, 6]. A famous example of such a matrix is 1 i+j n the Hilbert matrix i+j 1 . Another important example is the Pascal matrix = i i,j=0. Both − P of these are knownh to be infinitelyi divisible and totally positive. In [11], Cholesky decomposition of P was given, that is, a lower triangular matrix L was obtained such that = LL∗. Let = [ 1 ], where β( , ) is the beta function. We call as the betaP matrix. By definition, B β(i,j) · · B Γ(i)Γ(j) β(i, j)= , Γ(i + j) where Γ( ) is the Gamma function. Thus · (i + j 1)! = − . B (i 1)!(j 1)! − − ∗The research of this author is supported by INSPIRE Faculty Award IFA-14-MA52 of DST, India, and by Early Career Research Award ECR/2018/001784 of SERB, India. Email: [email protected] †Email: [email protected] ‡Email: [email protected] 1 Note that the entries of this matrix look similar to those of . Using the infinite divisibility and total positivity of , we show that is infinitely divisible and totallyP positive, respectively. We also compute the CholeskyP decomposition ofB . B Let Ei,j be the matrix whose (i, j)th entry is 1 and others are zero. For any complex numbers s, t, let Li(s)= I + sEi,i 1 and Uj (t)= I + tEj 1,j , − − where 2 i, j n. Matrices of the form Li(s) or Uj (t) are called elementary bidiagonal matrices. For ≤ ≤ a vector (d , d ,...,dn), let diag([di]) denote the diagonal matrix diag(d ,...,dn). An n n matrix is 1 2 1 × totally positive if and only if it can be written as (Ln (lk) Ln 1 (lk 1) L2 (lk n+2)) − − ··· − (Ln (lk n+1) Ln 1 (lk n) L3 (lk 2n+4)) (Ln (l1))D(Un (u1))(Un 1 (u2) Un (u3)) (U2 (uk n+2) − − − ··· −n ··· − ··· − ··· Un 1 (uk 1) Un (uk)), where k = , li,uj > 0 for all i, j 1, 2,...,k and D = diag[(di)] is a di- − − 2 ∈ { } agonal matrix with all di > 0 [12, Corollary 2.2.3]. This particular factorization (with li,uj 0) is called successive elementary bidiagonal (SEB) factorization or Neville factorization. We obtain≥ the 1 SEB factorization for β(i,j) explicitly. Let (x)0 = 1 and forh a positivei integer k, let (x)k = x(x 1)(x 2) (x k +1). The Stirling numbers of first kind s(n, k) [9, p. 213] and the Stirling numbers− of second− ··· kind−S(n, k) [9, p. 207] are respectively defined as n k (x)n = s(n, k)x k X=0 and n n x = S(n, k)(x)k. k X=0 The unsigned Stirling matrix of first kind s = [sij ] is defined as i j ( 1) − s(i, j) if i j, sij = − ≥ (0 otherwise. The Stirling matrix of second kind = [ ij ] is defined as S S S(i, j) if i j, ij = ≥ S (0 otherwise. It is a well known fact that s and are totally nonnegative (see for example [13]). We consider the S matrices ss∗ and ∗ and call them symmetrized unsigned Stirling matrix of first kind and symmetrized Stirling matrix ofSS second kind, respectively. By definition, these are positive semidefinite and totally nonnegative. We show that both these matrices are in fact totally positive. Another matrix that we consider is formed by the well known Bell numbers. The sum w(n) = n k=0 S(n, k) is the number of partitions of a set of n objects and is known as the nth Bell number [9, B w n 1 n 1 p. 210]. Consider the matrix = [ (i + j)]i,j−=0. Let X = (xij )i,j−=0 be the lower triangular matrix definedP recursively by x00 =1, x0j = 0 for j > 0, and xij = xi 1,j 1 + (j + 1)xi 1,j + (j + 1)xi 1,j+1 for i 1. − − − − ≥ (Here xi, 1 = 0 for every i and xin = 0 for i = 0,...,n 2.) This is known as the Bell triangle [8]. − − n 1 − Lemma 2 in [1] gives the Cholesky decomposition of B as LL∗, where L = X diag √i! . It is i=0 B n 1 B h i also shown in [1] that det = i=0− i!. It is a known fact that is totally nonnegative [18]. We show that B is totally positive. We also show that B is infinitely divisible only upto order 4. In Section 2 we give our resultsQ for the beta matrix, namely, its infinite divisibility and total positivity, 1 r its Cholesky decomposition, its determinant, − , and its SEB factorization. We show that ◦ is in B B 2 fact totally positive for all r> 0. We end the discussion on the beta matrix by proving that for positive 1 1 real numbers λ1 < < λn and µ1 < <µn, is an infinitely divisible matrix and ··· ··· β(λi,λj ) β(λi,µj ) is a totally positive matrix. In section 3, we proveh that symmetrizedi Stirling matrices and B areh totallyi positive. For the first kind, we give the SEB factorization for s. For the second kind, we show that is triangular totally positive [12, p. 3]. We also show that B is infinitely divisible if and only if its orderS is less than or equal to 4. 2 The beta matrix The infinite divisibility and total positivity of are easy consequences of the corresponding results for . For 1 i, j n, let A(i, j) denote the submatrixB of A obtained by deleting ith row and jth column fromP A. Each≤ A≤(i,i) is infinitely divisible, if A is infinitely divisible, and each A(i, j) is totally positive, if A is totally positive. Theorem 2.1. The matrix = 1 is infinitely divisible and totally positive. B β(i,j) h i Proof. By definition, 1 (i + j 1)! ij(i + j)! 1 i + j = − = = . (1) β(i, j) (i 1)!(j 1)! (i + j)i!j! 1 + 1 i − − i j 1 Thus = C (1, 1), where C = 1 + 1 is a Cauchy matrix. Both C and (1, 1) are infinitely divisible B ◦P i j P [3]. Since Hadamard product of infinitelyh i divisible matrices is infinitely divisible, we get is infinitely divisible. B Again, note that 1 (i + j 1)! (i + j 1)! (i 1)+ j = − = j − = − j. (2) β(i, j) (i 1)!(j 1)! (i 1)!(j!) i 1 − − − − So is the product of the totally positive matrix (n +1, 1) with the positive diagonal matrix diag([i]). HenceB is totally positive. P B The below remarks were suggested by the anonymous referee. r r Remark 2.2. Since for each r> 0, C◦ and ◦ are positive definite (see p. 183 of [4]), (1) gives that r n P r ◦ is positive definite. Let = [(i + j)!]i,j=0. Then is a Hankel matrix. Since ◦ is congruent to B r G r n rG G r ◦ via the positive diagonal matrix diag([i! ])i=0, ◦ is positive definite. The matrix ◦ (n +1, 1) = P r r G r G r [(i + j 1)! ] is congruent to ◦ , via the positive diagonal matrix diag [(i 1)! ]. So ◦ (n +1, 1) is − B r − G r also positive definite. Hence ◦ is totally positive, by Theorem 4.4 in [19]. This shows that ◦ is r G P totally positive. By (2), ◦ is also totally positive. B Remark 2.3. Another proof for infinite divisibility of can be given as follows.