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If I have that $A=PUP^{-1}$ where $U$ is upper triangular, how can I express A as $SLS^{-1}$ where $L$ is lower triangular? Is there a way to do this?

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Yes. You know that $P^{-1}AP=U$, where $U$ is upper triangular. Let $J$ be the matrix where the entries of the diagonal which is not the main one are all equal to $1$, whereas all other entries are equal to $0$. Then $J^{-1}UJ$ is lower diagonal; lets call it $L$. Then$$J^{-1}P^{-1}APJ=L$$and therefore$$A=(PJ)L(PJ)^{-1}.$$

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You can transform $U$ to be lower triangular by conjugating with the matrix having $1$'s down the diagonal top right to bottom left.

If that matrix is called $T$, then $S=PT$.

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  • $\begingroup$ I was thinking about doing that... The think is that I am trying to cause the change using the $S$ and $S^{-1}$. Is that possible? $\endgroup$ – MathIsHard Sep 11 '17 at 22:36
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    $\begingroup$ @Math4Life I think it is probably impossible to do just with $S$ in every case. $\endgroup$ – rschwieb Sep 11 '17 at 22:39

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