Customizable triangular factorizations of matrices

Abstract

Customizable triangular factorizations of matrices find their applications in computer graphics and lossless transform coding. In this paper, we prove that any N × N nonsingular matrix A can be factorized into 3 triangular matrices, A= PLUS , where P is a permutation matrix, L is a unit lower triangular matrix, U is an upper triangular matrix of which the diagonal entries are customizable and can be given by all means as long as its determinant is equal to that of A up to a possible sign adjustment, and S is a unit lower triangular matrix of which all but N−1 off-diagonal elements are set zeros and the positions of those N−1 elements are also flexibly customizable, such as a single-row, a single-column, a bidiagonal matrix or other specially patterned matrices. A pseudo-permutation matrix, which is a simple unit upper triangular matrix with off-diagonal elements being 0, 1 or −1, can take the role of the permutation matrix P as well. In some cases, P may be the identity matrix. Besides PLUS, a customizable factorization also has other alternatives, LUSP, PSUL or SULP for lower S , and PULS, ULSP, PSLU, SLUP for upper S .