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The payoff: principal axes, conics, and beyond R

When the form is symmetric, the spectral theorem upgrades congruence to a rotation, geometry snaps into focus, and conics and quadrics classify themselves. We close by stepping past the reals to Hermitian and symplectic forms.

The principal axis theorem

Completing the square diagonalizes by any invertible P, which distorts lengths and angles. The principal axis theorem says you can do better for a real symmetric form: there is an orthogonal P (a rotation/reflection, P^T = P^-1) with P^T A P diagonal. Because P is orthogonal, congruence and similarity coincide, so the diagonal entries are exactly the eigenvalues of A.

Classifying conics and quadrics

Now the geometric payoff. A general conic a x^2 + b xy + c y^2 + ... = 1 is q(x) plus linear terms; the principal axis theorem rotates away the cross-term, completing-the-square shifts away the linear terms, and the curve appears in standard position. The signature of the quadratic part then dictates the shape — no plotting required.

[[conic-classification]] in R^2 by the signature of the quadratic part:

  signature (2, 0)  both eigenvalues same sign  -> ELLIPSE
  signature (1, 1)  eigenvalues opposite signs  -> HYPERBOLA
  signature (1, 0)  one zero eigenvalue         -> PARABOLA (degenerate axis)

Worked example:  5 x^2 + 4 xy + 5 y^2 = 9
  A = [5, 2; 2, 5]   eigenvalues  3 and 7   (both > 0)
  rotate to principal axes:   3 u^2 + 7 v^2 = 9
  signature (2, 0)  ->  an ELLIPSE, semi-axes sqrt(9/3), sqrt(9/7).
The signature of the quadratic part classifies the conic with no graphing.

The same machine runs in any dimension. In R^3 the classification of quadrics — ellipsoids, hyperboloids of one or two sheets, paraboloids, cones — is read off the signature of a 3x3 symmetric matrix. One invariant triple (p, q, z), computed once, names a surface you would otherwise struggle to picture.

Beyond the reals: Hermitian and symplectic forms

Over C, x^T A x can be complex and useless for measuring length, so we conjugate one slot: a Hermitian form is linear in one argument and conjugate-linear in the other, with A satisfying A* = A (conjugate transpose). Its values q(x) = x* A x are always real, it has a real signature, and its own spectral theorem — Hermitian matrices are unitarily diagonalizable with real eigenvalues. This is the form underlying quantum mechanics.

And the skew thread from Guide 2 reaches its destination: a nondegenerate alternating form is a symplectic form, and inertia has a startling analogue — all symplectic forms of a given even dimension are congruent. There is exactly one, with standard matrix [0, I; -I, 0]; there is no signature, no shape to classify, because there is only one shape. This is the rigid geometry beneath Hamiltonian dynamics.