We call two square matrices A and B (over a ring) pseudo-similar, if matrices
X
,
X
−
,
X
=
X,{X^ - },{X^ = }
exist, such that
X
−
A
X
=
B
,
X
B
X
=
A
,
X
X
−
X
=
X
{X^ - }AX = B,XB{X^ = }A,X{X^ - }X = X
and
X
X
=
X
=
X
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.