The algebra of elimination

Abstract

Elimination with only the necessary row exchanges will produce the triangular factorization A = LPU, with the (unique) permutation P in the middle. The entries in L are reordered in comparison with the more familiar A = PLU (where P is not unique). Elimination with three other starting points 1, n and n, n and n, 1 produces three more factorizations of A, including the Wiener–Hopf form UPL and Bruhat’s U1πU2 with two upper triangular factors. All these starting points are useless for doubly infinite matrices. The matrix has no first or last entry. When A is banded and invertible, we look for a new way to establish A = LPU. The key is to locate the pivot rows (we also find the main diagonal of A). LPU was previously known in the periodic (block Toeplitz) case A(i,j)=A(i+b,j+b), by factoring a matrix polynomial.