I learned that given a matrix $A$, we can apply LU decomposition to get $A=LU$, where $L$ is lower triangular and $U$ is upper triangular. Further, if $A$ is symmetric (or Hermitian for complex $A$), then we can apply Symmetric Doolittle Factorization to get $LU=LDL^T$ as follows:
$$
\begin{align*}
LU&=A\\
&=A^T\\
&=U^TL^T\\
U&=L^{-1}U^TL^T\\
&=DL^T &\rightarrow\text{why}\,L^{-1}U^T=D?\\
LU&=LDL^T
\end{align*}
$$
But I do not understand why $L^{-1}U^T=D$, from what I understand, the inverse of a lower triangular is also a lower triangular, and the transpose of an upper triangular is lower triangular, so $L^{-1}U^T$ is the matrix multiplication of two lower triangular matrices, which will result in a lower triangular matrix, but I do not see how it is going to be a diagonal matrix. May someone help to explain it to me? Thanks!
Best Answer
We define $D := L^{-1} U^T$. Therefore, as you wrote, $$ U = L^{-1} U^T L^T = D L^T \Rightarrow\\ U = D L^T \Rightarrow\\ U (L^{-1})^T = D \Rightarrow\\ D^T = L^{-1} U^T $$ From the equation above one can conclude, that $D^T = D$. Given that $D$ is lower triangular and symmetric we derive, that $D$ is diagonal.