LU-decomposition of a matrix with entries of different kinds (Q788069)

Given a matrix A with entries in a field F, which is extension field of a field K, that has a decomposition \(A=Q+T\) where Q is a matrix with entries in K and all nonzero entries of T are algebraically independent transcendentals over K. It is shown that if det \(A\in K\backslash \{0\}\) then there exist permutation matrices \(P_ r,P_ c\) and an LU- decomposition of \(P_ r^ TAP_ c\) where L,U have the following properties: The elements of L are polynomials of degree at most 1 with coefficients in K in the nonzero entries of T and the entries of U are in K. It is indicated how to find the suitable permutations to achieve this decomposition. The problem of finding an above factorization occured in large-scale systems analysis.