Principal axis theorem

Statement

Let $A$ be a real symmetric $n \times n$ matrix, and let $Q(x) = x^T A x$ for $x \in \R^n$. Then there are an orthogonal matrix $P$ and real numbers $\lambda_1, \dots, \lambda_n$ such that, for $x = P y$, $$Q(x) = \lambda_1 y_1^2 + \lambda_2 y_2^2 + \cdots + \lambda_n y_n^2.$$ The numbers $\lambda_i$ are the eigenvalues of $A$, and the columns of $P$ are orthonormal eigenvectors of $A$.

Remarks

It follows from the spectral theorem $A = P D P^T$ with $D$ diagonal. The lines spanned by the columns of $P$ are the principal axes of $Q$. For example, $Q(x) = 5x_1^2 + 4x_1 x_2 + 2x_2^2$ has the eigenvalues $6$ and $1$, so it becomes $6y_1^2 + y_2^2$. So the curve $Q(x) = 1$ is an ellipse, with semi-axes $1/\sqrt 6$ and $1$.

Explanations

L1 human

Each real quadratic form becomes a sum of squares after a rotation of the axes, possibly with a reflection. The new axes are the principal axes of the form. This is how we find the shape of an ellipse or a hyperbola from its equation.