ALGEBRA 2026-05-05

Spectral Theorem

Theorem

(Spectral Theorem (Real Symmetric Matrices))

If $A\in\mathbb{R}^{n\times n}$ satisfies $A=A^\top$ , then there exists an orthogonal matrix $Q$ and a diagonal matrix $D$ such that

A=QDQ^\top.

The diagonal entries of $D$ are the eigenvalues of $A$ , all of which are real, and the columns of $Q$ are orthonormal eigenvectors.

Remark

(Geometric Interpretation)

The spectral theorem says that every symmetric linear operator on $\mathbb{R}^n$ is simply a rescaling along $n$ mutually perpendicular axes — the eigenvector directions.