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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.

Thanks in advance.

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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*.2012-07-02
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    I think congruence transformation and isometry are equivalent, see [wiki](http://en.wikipedia.org/wiki/Congruence_%28geometry%29), and I don't think similarity transformation requires the bases to be orthogonal.2012-07-02
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    Are you talking about transformations of the underlying space, or the equivalence relations among matrices called "similarity" and "congruence"?2012-07-02
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    What do you mean by a congruence transformation?2012-07-03
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    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$^{-1}$.2012-07-03
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    So just to clarify further in response to Dr. Magidin's question, I believe the transformations to which I am referring are the equivalence relations on matrices called "similarity" and "congruence".2012-07-03

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