Parameterization of totally positive matrices

4.ii. Parameterization of totally positive matrices

There is a very interesting a detailed theory of parameterizations of totally positive matrices TP [BFZ]. For our purposes, we select one particular parameterization. Let a_i,j for i=1, 2, ..., m+p-1 and j < i be positive numbers. Let D₁, D₂, ..., D_m+p-1 be matrices with 1's on their diagonals and with the first upper diagonal of D_i the sequence (a_i,1, a_i,2, ..., a_i,i-1, 0, ..., 0). Then the product D₁D₂... D_m+p-1 is a matrix in TP, and the positive numbers a_i,j parameterize TP.