Symmetric Matrices

A matrix is symmetric if $A = A^T$.

Spectral Theorem

Every symmetric matrix can be diagonalized:

$$A = Q\Lambda Q^T$$

Where $Q$ is orthogonal ($Q^TQ = I$).

Key Properties

1. All eigenvalues are real
2. Eigenvectors are orthogonal (can choose orthonormal)
3. Always diagonalizable (even with repeated eigenvalues)
4. $A = Q\Lambda Q^T$ (not $S\Lambda S^{-1}$ — no inverse needed!)

Why Symmetric Is Special

• No complex eigenvalues
• No defective (non-diagonalizable) cases
• Orthogonal eigenvectors = numerically stable
• Most matrices in practice are symmetric (or close)