Positivity properties of some special matrices

Abstract

It is shown that for positive real numbers 0 < λ 1 < · · · < λn, 1 β(λ i ,λ j ) , where β(·, ·) denotes the beta function, is infinitely divisible and totally positive. For 1 β(i,j) , 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.

High relative accuracy with some special matrices related to Γ and β functions

“…for r ≤ i, j ≤ n. So, taking into account the previous formula and that in this first step of the NE a (r) ij = a (r−1) ij for i = 1, … , r − 1 and j = i, … , n, we conclude that (13) holds for r and the induction follows. By (13) for r = i = j we have that…”

“…In Section 3, we provide, for a family of matrices using the function $$$ \Gamma $$$, their bidiagonal factorization and we prove that it (and so the remaining algebraic computations mentioned above) can be performed with HRA when $$$ \Gamma $$$ is defined on integers. In Section 4, we prove the strict total positivity of a more general family of matrices than that considered in Reference 13 that used the $$$ \beta $$$ function. Moreover, we provide their bidiagonal factorizations, and prove that the algebraic computations mentioned above can be performed with HRA when these matrices are defined on integers.…”

“…In Section 2, we introduce basic notations and auxiliary results related with the bidiagonal factorization of a nonsingular TP matrix and the HRA. In Reference 13, it was proved that the matrix $$$ {\left(\frac{1}{\beta \left(i,j\right)}\right)}_{1\le i,j\le n} $$$ is STP and its bidiagonal factorization was provided, and it was also proved the strict total positivity of some matrices using the function $$$ \Gamma $$$. In Section 3, we provide, for a family of matrices using the function $$$ \Gamma $$$, their bidiagonal factorization and we prove that it (and so the remaining algebraic computations mentioned above) can be performed with HRA when $$$ \Gamma $$$ is defined on integers.…”

On computing the symplectic LLT factorization

“…B is symmetric totally positive matrix of integer. More detailed information related to beta matrix can be found in [4] and [6].…”

Positivity of Hadamard powers of a few band matrices

“…Some basic examples of ID matrices are nonnegative PSD matrices of order 2 and diagonal matrices with nonnegative diagonal entries. For more examples and results on ID matrices, see [1,3,6,10]. In our next theorem, we give a characterization for the matrix T to be infinitely divisible.…”