Characterization of the existence of an \(L\)-\(U\) factorization

For the first time, a characterization is given of the circumstances under which an n-by-n matrix over a field has an \(L\)-\(U\) factorization. This is in terms of a comparison of ranks of the leading k-by-k principal submatrix to the rank of the first k columns and first k rows. Known results about special types of \(L\)-\(U\) factorizations follow as do some new results about near \(L\)-\(U\) factorization when a conventional \(L\)-\(U\) factorization does not exist. The proof allows explicit construction of an \(L\)-\(U\) factorization when one exists.