# Name for 'flipped' triangular matrix

My understanding, albeit limited, is that the Cholesky decomposition of a symmetric positive definite square matrix is:

$A = L L^{*}$

where $L^{*}$ is the conjugate transpose of $L$ and $L$ is a lower triangular matrix known as the Cholesky factor:

$\begin{pmatrix} \# & 0 & 0 & 0 \\ \# & \# & 0 & 0 \\ \# & \# & \# & 0\\ \# & \# & \# & \#\end{pmatrix}$

$A = U^{*}U$

where $U$ is:

$\begin{pmatrix} 0 & 0 & 0 & \# \\ 0 & 0 & \# & \# \\ 0 & \# & \# & \#\\ \# & \# & \# & \#\end{pmatrix}$

And it would seem that

$U = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\end{pmatrix}L^{*}$

First question: the author claims that U is a Cholesky factor but $U \neq L$. So, what is it?

And if $U$ is not a triangular matrix (as far as I can tell from various sources online), is there a name for it?

• I think I've heard the term "anti-triangular" for this kind of matrices. – Calle Apr 2 '15 at 14:29