Let $S$ be a lower triangular matrix, $T$ an upper triangular matrix and $D$ a diagonal matrix. Suppose all of them are invertible, i.e., all diagonal elements are non-zero.
If $SDT$ is symmetric, then is $ST$ symmetric?
(Sorry for the previous question.)
For an example:
$$S=\left(\begin{matrix}2&0&0\\1&5&0\\3&-1&6\end{matrix}\right),\qquad D=\left(\begin{matrix}1&0&0\\0&2&0\\0&0&3\end{matrix}\right), \qquad T=\left(\begin{matrix}2&1&3\\0&-5&1\\0&0&2\end{matrix}\right).$$
Since $SDT$ is symmetric, we have $SDT=T^TDS^T$ and hence $DTS^{-T}=S^{-1}T^TD$. Denoting $X=TS^{-T}$, we have $$\tag{1} DX=X^TD. $$ Note that $X$ is upper triangular because $T$ and $S^{-T}$ are upper triangular. However, (1) implies that $X$ is also lower triangular and consequently, $X$ is diagonal. It means that $X=TS^{-T}=C$, where $C$ is a non-singular diagonal matrix. Therefore, $S$ and $T$ are related by $T=CS^T$ and it follows that $ST=SCS^T$ is symmetric.
