I'm seeking for a transformation matrix $T\in\mathbb{R}^{n\times n}$ such that a given matrix $A\in\mathbb{R}^{n\times n}$ satisfies the following identity:
$A = T^{-1}UT$,
where $U\in\mathbb{R}^{n\times n}$ is an upper-triangular matrix. How do I obtain such matrices $T$ and $U$? Furthermore, would there be a similar transformation for a block matrix? For example, for a block matrix $A\in\mathbb{R}^{(m+n)\times(m+n)}$,
$\displaystyle A \equiv \left( \begin{array}{c : c} L & L_Q \\ \hdashline 0 & Q \end{array} \right) = T^{-1} \left( \begin{array}{c : c} \tilde{L} & 0 \\ \hdashline \tilde{L}_Q & \tilde{Q} \end{array} \right) T $,
where $L, \tilde{L} \in \mathbb{R}^{m\times m}$, $L_Q\in \mathbb{R}^{m\times n}$, $\tilde{L}_Q \in \mathbb{R}^{n\times m}$, and $Q, \tilde{Q} \in \mathbb{R}^{n\times n}$.