Minimum deviation, quasi-LU factorization of nonsingular matrices

Not all matrices enjoy the existence of an LU factorization. For those that do not, a number of “repairs” are possible. For nonsingular matrices we offer here a permutation-free repair in which the matrix is factored , with and collectively as near as possible to lower and upper triangular (in a natural sense defined herein). Such factorization is not generally unique in any sense. In the process, we investigate further the structure of matrices without LU factorization and permutations that produce an LU factorization.