What do you call this equivalence relation? $A \simeq B$ if $A = P^t BP$ for some invertible matrix $P$ If $A, B$ are square matrices with coefficients in some ring, we say that $A$ is similar to $B$ if $A = PBP^{-1}$ for some invertible matrix $P$.  Similar matrices represent the same linear operator with respect to different bases.
Now another equivalence relation is to set $A \simeq B$ is $A = P^tAP$ for some invertible matrix $P$, where $P^t$ denotes the transpose.  Two matrices are equivalent if they give the same bilinear form with respect to different bases.
What is this equivalence relation usually called? In Jacobson's Basic Algebra, he says that $A$ and $B$ are cogredient.  But I haven't seen this terminology elsewhere.
 A: These names/terminologies are common, but completely opaque and non-descriptive, indeed. Much like saying a subgroup is "normal" suggests nothing at all about the feature it apparently has.
The operational point is that $A \to PAP^{-1}$ re-expresses the linear map given by $A$ under the change of basis given by $P$. Nowadays, the specific matrix operation might be called conjugation of $A$ by $P$. In any case, it is the way to see the matrix of a linear transformation under a change of basis.
The change $A \to P^\top A P$ is change of coordinates when using a matrix $A$ to represent a _quadratic_form_ $v\to v^\top Av$, for vectors $v$.
So, apart from the common-but-unhelpful labels, these are change-of-coordinates for two different uses of a matrix $A$.
A: According to Horn & Johnson, Matrix Analysis (p. 281): Such matrices are called congruent or $\,^T\!$congruent ("tee-congruent"), and if $B = S^*AS$ for some invertible $S$ (where $S^*$ denotes conjugate transpose), then $A$ and $B$ are called *congruent ("star-congruent") or conjunctive.
The terms $\,^T\!$congruent and *congruent are unambiguous and easy to remember. I don't like cogredient here: it's unusual and it's been used to mean other things. In Berman & Plemmons, Nonnegative Matrices in the Mathematical Sciences (p. 27), two matrices are called cogredient if they're related by conjugation via a permutation matrix: $A = P^t B P$ for some permutation matrix $P$. In keeping with the theme, maybe we should call such matrices $\,^\pi\!$congruent?
