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  "body": "\n## Statement\n\nLet $A$ be a real symmetric $n \\times n$ matrix, and let $Q(x) = x^T A x$ for $x \\in \\R^n$.\nThen there are an [[orthogonal matrix|0W02QESPJQ]] $P$ and real numbers $\\lambda_1, \\dots, \\lambda_n$ such that, for $x = P y$,\n$$Q(x) = \\lambda_1 y_1^2 + \\lambda_2 y_2^2 + \\cdots + \\lambda_n y_n^2.$$\nThe numbers $\\lambda_i$ are the eigenvalues of $A$, and the columns of $P$ are orthonormal eigenvectors of $A$.\n\n## Remarks\n\nIt follows from the [[spectral theorem|0DGZBQWZWN]] $A = P D P^T$ with $D$ diagonal.\nThe lines spanned by the columns of $P$ are the *principal axes* of $Q$.\nFor example, $Q(x) = 5x_1^2 + 4x_1 x_2 + 2x_2^2$ has the eigenvalues $6$ and $1$, so it becomes $6y_1^2 + y_2^2$.\nSo the curve $Q(x) = 1$ is an ellipse, with semi-axes $1/\\sqrt 6$ and $1$.\n\n## Explanations\n\n### L1 human\n\nEach real quadratic form becomes a sum of squares after a rotation of the axes, possibly with a reflection.\nThe new axes are the principal axes of the form.\nThis is how we find the shape of an ellipse or a hyperbola from its equation.\n",
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