Inherited Matrix Entries: $LU$ Factorizations

Abstract

For an n-by-n matrix $A = [ a_{ij} ]$ which has a unique unit $LU$ factorization $A = LU$ with $U = [ u_{ij} ]$, combinatorial circumstances are determined under which $u_{ij} = a_{ij} $ for a given pair $i\leqq j$ or for all $i < j$ (or all $i\leqq j$). Analogous results are stated for other triangular factorizations and for the $LU$ factorization of a principal submatrix of A. The relationship of the results to Gaussian elimination and sparse matrix analysis is discussed.