Bidiagonal decompositions and total positivity of some special matrices

The matrix S = [1 + x i y j ] ni,j =1 , 0 < x 1 < · · · < x n , 0 < y 1 < · · · < y n , has gained importance lately due to its role in powers preserving total nonnegativity. We give an explicit decomposition of S in terms of elementary bidiagonal matrices, which is analogous to the Neville decomposition . We give a bidiagonal decomposition of S ◦ m = [(1 + x i y j ) m ] for positive integers 1 ≤ m ≤ n − 1. We also explore the total positivity of Hadamard powers of another important class of matrices called mean matrices .