# Difference between congruence and similarity transformations

I am trying to understand the difference between a "congruence" or "similarity" transformation for two $n \times n$ matrices (which for the sake of simplicity, we'll assume are real). From what I can glean, a similarity transformation represents a change of basis from one orthogonal basis in $\mathbf{R}^n$ to another. My understanding is that a congruence transformation is an isometry, and so, it seems it would represent some geometrical operation like a (rigid) rotation, reflection, etc which preserves angles ad distances (but not necessarily orientation).

If someone can tell me if this is correct or correct any mistakes in my interpretation, I'd be most appreciative.

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Congruence transformation is indeed isometry. The description in terms of changing from one orthogonal basis to another is not quite right, since for example the matrix with $2$ down the main diagonal, $0$ elsewhere doubles distances. Maybe you meant orthonormal. –  André Nicolas Jul 2 '12 at 20:30
I think congruence transformation and isometry are equivalent, see wiki, and I don't think similarity transformation requires the bases to be orthogonal. –  chaohuang Jul 2 '12 at 21:07
Are you talking about transformations of the underlying space, or the equivalence relations among matrices called "similarity" and "congruence"? –  Arturo Magidin Jul 2 '12 at 21:56
What do you mean by a congruence transformation? –  Qiaochu Yuan Jul 3 '12 at 0:10
The definitions I am using are as follows: Let A and B be two real nxn matrices. A = P B Q (where P and Q are both nonsingular) is said to be a congruence transformation if P=Q$^{T}$ and is said to be a similarity transformation if P = Q$^{-1}$. –  Matt Brenneman Jul 3 '12 at 2:59

From what I can glean, a similarity transformation represents a change of basis from one orthogonal basis in $\mathbb R^n$ to another.
This is incorrect. A similarity transformation $A\mapsto P^{-1}AP$ indeed amounts to a change of basis (in the sense that both matrices represent the same linear map in different bases), but neither the original nor the new basis need be orthogonal. They can be any two bases.
This is incorrect, or at least imprecise. Matrix congruence transformation $A\mapsto P^T AP$ also amounts to a change of basis, but in the sense that both matrices represent the same quadratic form in different bases. I think you confused the matrix congruence with the notion of congruence of sets in geometry. These are two different things with the same name (and there are more).