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# The spectral theorem for symmetric matrices

Put content here.**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:
$$A = Q D Q^T$$
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Equivalently:
- $A$ has an orthonormal basis of eigenvectors
- All eigenvalues of $A$ are real
- Eigenspaces corresponding to distinct eigenvalues are orthogonal
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**Example.** For $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$:
$$A = \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{pmatrix} \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{pmatrix}^T$$

# Parents

* Symmetric matrices