= Solution
A real <symmetric matrix> has a real orthonormal eigenbasis. Put those <eigenvectors> into an <orthogonal matrix> $Q$, so $Q^TAQ=\operatorname{diag}(a_1,\ldots,a_n)$. With the change of coordinates $x=Qx'$, or $x'=Q^Tx$,
$$
\boxed{\sum_{ij}A_{ij}x_ix_j=x^TAx=x'^T(Q^TAQ)x'=\sum_i a_i(x_i')^2.}
$$
Thus the coefficients are the <eigenvalues> and the new coordinate directions are the <eigenvectors>. This <orthogonal diagonalization> preserves Euclidean lengths and angles while removing cross terms from the <quadratic form>.
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