9 Quadratic Forms

A linear form is a sum of terms in \(x_i\); a quadratic form is a sum of terms in \(x_ix_j\). Expressions of this kind describe distances, energies, variances and conic sections, and the question in every case is what the expression looks like once the axes are chosen well.

Every quadratic form can be written \(Q(x)=x^tAx\) with \(A\) symmetric, which hands the problem to Section 8. The spectral theorem says a real symmetric matrix is diagonalisable by an orthogonal matrix, so the change of variable required is a rotation — it preserves lengths and angles, and therefore does not distort the object being described.

In the rotated coordinates the cross terms vanish and the form becomes \(\lambda _1y_1^2+\cdots +\lambda _ny_n^2\). The signs of the eigenvalues then classify it, which for two variables is precisely the classification of conic sections into ellipses, hyperbolas and parabolas.

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