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 ai,j for i=1, 2, ..., m+p-1 and
j < i be positive numbers.
Let D1, D2, ...,
Dm+p-1 be matrices
with 1's on their diagonals and with the first upper diagonal of
Di the sequence
(ai,1, ai,2, ...,
ai,i-1, 0, ..., 0).
Then the product
D1D2...
Dm+p-1 is a matrix in
TP, and the positive numbers
ai,j parameterize
TP.
Unfortunately, the total positivity conjecture has been found to be false, and so this is
no longer relevant.