principal axis theorem

The principal axis theorem says that any real quadratic form can be diagonalized by an ORTHOGONAL change of variables — a rigid rotation (and possibly reflection) of the coordinate axes, no stretching. There is an orthonormal basis of eigenvectors of the symmetric matrix B in which Q becomes lambda_1 y_1^2 + ... + lambda_n y_n^2, and this time the coefficients ARE the eigenvalues of B.

This is the spectral theorem in disguise. Because B is real symmetric, the spectral theorem gives an orthogonal matrix P (so P^T = P^-1) with P^T B P = diag(lambda_i). Here the magic is that P^T B P and P^-1 B P coincide — congruence and similarity agree exactly when P is orthogonal. So this single rigid change of basis simultaneously diagonalizes the operator and the form.

The geometric payoff is the 'principal axes' of a conic or quadric. For an ellipse x^T B x = 1, the eigenvectors of B point along the major and minor axes, and the eigenvalues set the axis lengths (semi-axis = 1/sqrt(lambda)). Rotating to the eigenbasis turns a tilted, cross-term-laden equation into the textbook form with the axes aligned to the coordinate directions.

Contrast this carefully with plain diagonalization-by-congruence. Completing the square uses an arbitrary invertible P and only fixes the SIGNS of the coefficients (signature); the principal axis theorem insists P be orthogonal and recovers the ACTUAL eigenvalues, preserving lengths and angles. You pay for the extra rigidity with the requirement that the field be the reals and the form be symmetric.