Source: cirosantilli/matrix-congruence-can-be-seen-as-the-change-of-basis-of-a-bilinear-form

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

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:
$$
C^T M C
$$

So, by taking $S = C^T$, 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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