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:
$\begin{pmatrix} \# & 0 & 0 & 0 \\ \# & \# & 0 & 0 \\ \# & \# & \# & 0\\ \# & \# & \# & \#\end{pmatrix}$
In a paper I've been reading about adaptive filters, I've come across a similar decomposition:
$A = U^{*}U$
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?