proof of a lemma

Lemma

If the symmetric matrices M and N define the same conic C, then
N = kM for some non-zero k.

Proof

Applying a projective transformation, we may as well assume that the conic is C0.
Suppose that M leads to the conic ax2+bxy+cy2+fzx+gyz+hz2 = 0.
Since X,Y,Z lie on the conic, we see that a = c = h = 0.
By the Parametrisation Theorem, any other point on C0 has the form (t,1-t,t(t-1)).
Thus, we must have bt(1-t)+ftt(t-1)+g(1-t)t(t-1) = 0 for all t. This means that the
polynomial must vanish identically. Comparing coefficients of t and t3, we see that
b = f = g. Thus M and N must be multiples of one another.

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