Pseudo-similarity for matrices over a field

We call two square matrices A and B (over a ring) pseudo-similar, if matrices $X,{X^ - },{X^ = }$ exist, such that ${X^ - }AX = B,XB{X^ = }A,X{X^ - }X = X$ and $X{X^ = }X = X$. We show that if A and B have the same dimension and if the ring is a field, then pseudo-similarity implies similarity, and hence that pseudo-similarity is an equivalence relation.