# Upper triangular matrix transform to lower triangular matrix

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?

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$$A=(PJ)L(PJ)^{-1}.$$
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 $S=PT$.
• 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? – MathIsHard Sep 11 '17 at 22:36
• @Math4Life I think it is probably impossible to do just with $S$ in every case. – rschwieb Sep 11 '17 at 22:39