The spectral theorem for symmetric matrices

Spectral Theorem for Real Symmetric Matrices. Every real symmetric matrix $A$ can be diagonalized by an orthogonal matrix. That is, there exists an orthogonal matrix $Q$ (with $Q^T Q = I$) and a diagonal matrix $D$ such that:
$$A = Q D Q^T$$

Equivalently:

• $A$ has an orthonormal basis of eigenvectors
• All eigenvalues of $A$ are real
• Eigenspaces corresponding to distinct eigenvalues are orthogonal

Example. For $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$:
$$A = \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{pmatrix} \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{pmatrix}^T$$