📐 Spectral Theorem (Real Symmetric Matrices)

Every real symmetric matrix is orthogonally diagonalizable: $A = QDQ^T$ where $Q$ is orthogonal and $D$ is diagonal.

Proof: **Step 1: Eigenvalues are real.** If $Av = \\lambda v$ with $v \\neq 0$:

$$\\bar{\\lambda} \\bar{v}^T v = \\bar{v}^T \\bar{\\lambda} v = \\bar{v}^T A v = (A^T \\bar{v})^T v = (A\\bar{v})^T v = \\lambda \\bar{v}^T v$$

Since $\\bar{v}^T v = \\|v\\|^2 > 0$, we have $\\bar{\\lambda} = \\lambda$.

**S...

From: Advanced Linear Algebra

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