Stemming from this question when actually searching for matrix similarities, having found this matrix equivalence:
$$\bf A = PBP$$ That is neither transpose nor inversion on either of the $\bf P$s. Just the same matrix from both left and right. I have not found any other properties for the $\bf P$s found but I am curious for any properties it could have and what use they could be of.
As I rarely have seen this type of equivalence before I got curious of where it could pop up when solving problems or making models in science and engineering. Do you have any exciting examples or stories?
EDIT Example of a $\bf P$ matrix:
$$\left[\begin{array}{rr} -1.09067057844122&-0.983363128995121\\ 0.0634758169341177&-1.12643972775094 \end{array}\right]$$
I have not managed to find any special structure or property it has yet.