Given a real n$\times$n matrix $A$, all diagonal entries are $0$, there are at most $(n-1)$ non-zero entries in $A$, and if $A(i,j)$ is non-zero, $A(j, i)$ is $0$. For example, a matrix $A$ is
\begin{bmatrix} 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0 & 1 & 0 & 0 & 0\\ 4 & 0 & 3 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ \end{bmatrix}
I want to find a permutation matrix $T$, such that $TAT^T$ is a strictly lower triangular matrix. If possible, I want the non-zero entries which are already located at the strictly lower triangular in matrix $A$ keep staying at their original row and column at much as possible.
For example, I want to move '2' in the above matrix $A$ into a strictly-lower triangular location, at the same time, I want to keep '$4, 1, 3$' at their original locations as much as possible.
What is the general algorithm to find such a permutation matrix $T$? Thank you all very much.