= Real spectral theorem
{title2=$A=Q\Lambda Q^T,\quad Q^TQ=I$}
The real case of the <finite-dimensional spectral theorem> says that a <self-adjoint operator> on a finite-dimensional real <inner product space> has a real <orthonormal eigenbasis>. In orthonormal coordinates this is the <spectral theorem for real symmetric matrices>, $A=Q\Lambda Q^T$. The worked <ellipse> under <finite-dimensional spectral theorem> shows why both <orthogonal diagonalization> and <orthogonality> matter: the new coordinates separate the <quadratic form> into squares without distorting <lengths> or <angles>.
A real <normal matrix> need not have real <eigenvalues>. For example, the quarter-turn $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ is an <orthogonal matrix>, but its <characteristic polynomial> is $z^2+1$ and its <eigenvalues> are $i,-i$. Over the <complex numbers> it has an <orthonormal eigenbasis>; over the <real numbers> it has no <eigenvector>. Self-adjointness supplies the reality conclusion in the real spectral theorem.
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