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If I have two matrices $A$ and $B$, where both have the same eigenvalues and $B$ is positive definite, is it possible to simultaneously triangularize them? So that there exists an invertible matrix $P$, s.t. $PAP^{-1}$ is triangular and $PBP^{-1}$ is triangular?

Actually, I want to do this with these two matrices:

$$A=\left( \begin{array}{ccc} \frac{3}{2} & -\frac{3}{2} & \frac{1}{2} \\ 0 & 2 & 0 \\ -1 & 0 & 0 \\ \end{array} \right),\quad B=\left( \begin{array}{ccc} 2 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & \frac{1}{2} \\ \end{array} \right).$$

If not, would it be possible to make them both block-diagonal?

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