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Is there a way to interpret "congruence " of matrices or linear maps? For example, "similar" maps can be interpreted as the same maps wrt different bases. Is there something like that corresponding to "congruent" maps?

Added: I have found on wikipedia that it means changing basis on the Gram matrix attached to a bilinear form, how is this different from the change of basis for normal linear maps? and why does using transposes work?

Thank you.

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1 Answer 1

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A matrix $A$ defines a bilinear map $\langle\cdot,\cdot\rangle_A:(v,w)\mapsto v^TAw$, where $v,w$ are column vectors. If $P$ is an invertible linear map it represents a change-of-basis, hence

$$\langle Pv,Pw\rangle_A=(Pv)^TA(Pw)=v^T(P^TAP)w=\langle v,w\rangle_{P^TAP}.$$

Thus congruent matrices represent the same bilinear map under changes of basis.

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