Effect of a change of basis on the matrix of a bilinear form

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Effect of a change of basis on the matrix of a bilinear form by Ciro Santilli 34 Updated 2024-12-15 +Created 1970-01-01
If is the change of basis matrix, then the matrix representation of a bilinear form that looked like:
(1)
then the matrix in the new basis is:
(2)
Sylvester's law of inertia then tells us that the number of positive, negative and 0 eigenvalues of both of those matrices is the same.
Proof: the value of a given bilinear form cannot change due to a change of basis, since the bilinear form is just a function, and does not depend on the choice of basis. The only thing that change is the matrix representation of the form. Therefore, we must have:
(3)
and in the new basis:
(4)
and so since:
(5)
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