= Solution
For a real <symmetric matrix> $A$, find its real <eigenvalues> from $\det(A-\lambda I)=0$, then solve $(A-\lambda I)v=0$ for the corresponding <eigenspaces>. Distinct <eigenspaces> are orthogonal. Within each repeated <eigenspace>, apply the <Gram-Schmidt process> and normalize to obtain an <orthonormal basis> of <eigenvectors>.
Put these vectors in the columns of $O$. The <spectral theorem for real symmetric matrices> guarantees a full basis, so $O^TO=I$ and
$$
\boxed{O^TAO=\operatorname{diag}(\lambda_1,\ldots,\lambda_n),\qquad A=O\operatorname{diag}(\lambda_1,\ldots,\lambda_n)O^T.}
$$
This is the <orthogonal diagonalization> of $A$ and the associated <orthogonal change of coordinates>.
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