DU decomposition

Question

My professor said that we can't express a square matrix as a product of upper triangular matrix and lower triangular although it can be expressed as a product of lower triangle matrix and an upper triangular matrix. I'm not convinced with her answer. Is she right or not?

Answer 1

It is in fact always possible. This can be seen by understanding that one can obtain a lower triangular matrix from an upper triangular one, and vice versa, by simply reversing the order of indexing.

Suppose $A$ is the matrix in question, and let $\tilde{A}$ be the same matrix but with indices reordered in reverse, in particular $\tilde{A} = PAP$ where $P$ is the permutation matrix that reverses the order of the indices; $P$ is a square matrix consisting of all zeroes except for $1$s on the anti-diagonal (the diagonal of the matrix that goes from bottom-left to upper-right). Then $\tilde{A}$ has an $LU$ decomposition. But now simply reverse the indices again, and this gives a decomposition of $A$ into $\tilde{U} \tilde{L}$, where $\tilde{U} = PUP$ and $\tilde{L} = PLP$, with $L,U$ being the $LU$ factors of $\tilde{A}$.