Multiple LU factorizations of a singular matrix
A general \(m\)-by-\(n\) matrix \(A=(A_{i,j})\) over the complex field \(\mathbb C\) has an LU factorization if it can be written as \(A=LU\) with \(L\) \(m\)-by-\(m\) lower triangular (\(L_{ij}=0\) whenever \(i < j\)) and \(U\) \(m\)-by-\(n\) upper triangular (\(U_{ij}=0\) whenever \(i > j\)). A nonsingular \(L\) can be normalized to have 1's on the diagonal and is called unit lower triangular. The authors describe all possible LU factorizations, with \(L\) nonsingular unit lower triangular, of any singular matrix \(A\) for which such a factorization exists. Any matrix \(A\) is factorized under left multiplication by unit lower triangular matrices. In this way a new canonical form, whose entries can be determined in terms of quotients of minors of \(A\), is introduced. This canonical form allows easily to characterize the matrices having an LU factorization. Formulae, in terms of quotients of minors of \(A\), the denominators of which are nonzero, are given for those entries of \(L\) and \(U\) that are uniquely determined.
scientific article; zbMATH DE number 5077724