Congruence of forms

Hi Professor Kim,

I was just wondering how you determine whether two quadratic forms q and r are congruent over R and C. I know that you proceed by finding their canonical forms by doing double elementary row and column operations, but what do you do next? Are they congruent if they have the same canonical form over R (or over C) OR is this the definition of being equivalent. Please help.

Thank you very much.

Reply:

Two forms q_1 and q_2 are congruent if any of their representing matrices A_1 and A_2 are congruent. Notice that this definition doesn’t depend on the choice of representing matrices. (Why not?) Of course, it suffices to diagonalize (in the sense of congruence, i.e., using double operations) A_1, A_2 and then check if the diagonal matrices are congruent. This can still be hard. But over R or C,

A_1 and A_2 are congruent if and only if their canonical forms are *equal*.

This is a very useful criterion, since we can trivially check two matrices for equality.

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