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by Ciro Santilli (@cirosantilli, 37)

Matrix congruence can be seen as the change of basis of a bilinear form

 ... Linear algebra Matrix Eigenvalues and eigenvectors Eigendecomposition of a matrix Sylvester's law of inertia Congruent matrix
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From effect of a change of basis on the matrix of a bilinear form, remember that a change of basis C modifies the matrix representation of a bilinear form as:
CTMC
(1)
So, by taking S=CT, we understand that two matrices being congruent means that they can both correspond to the same bilinear form in different bases.

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