94. Spectral Theorem

The Spectral Theorem says that every real symmetric matrix is orthogonally diagonalizable:

where:

• $𝐷$ is a diagonal matrix whose entries are the (always real) eigenvalues of $𝐴$
• $𝑄$ is an orthogonal matrix whose columns are an orthonormal basis of eigenvectors of $𝐴$

Equivalently, ${𝑄}^{𝑇}𝑄=𝐼$, and the columns of $𝑄$ are orthonormal eigenvectors.

94.1. What it guarantees

For a real symmetric matrix $𝐴$:

1. All eigenvalues are real (no complex eigenvalues, even if $𝐴$’s entries don’t suggest it)
2. Eigenvectors for distinct eigenvalues are orthogonal (automatic)
3. Eigenvectors for repeated eigenvalues can be chosen orthonormal (within that eigenspace, via Gram–Schmidt)
4. The full set of $𝑛$ orthonormal eigenvectors forms a basis of ${ℝ}^{𝑛}$

The trio (1)+(2)+(3) is what makes the theorem work. No real symmetric matrix is missing eigenvectors.

94.2. Spectral expansion

Write $𝑄=\left[\begin{array}{cccc}{𝑞}_1& {𝑞}_2& \dots & {𝑞}_{𝑛}\end{array}\right]$ and $𝐷=\text{diag}\left({𝜆}_1,\dots ,{𝜆}_{𝑛}\right)$. Then:

The matrix decomposes into a sum of rank-1 spectral projectors ${𝑞}_{𝑖}{𝑞}_{𝑖}^{𝑇}$, each scaled by an eigenvalue. This is the spectral decomposition.

94.3. Generalizations

• Complex Hermitian matrices (${𝐴}^{\ast }=𝐴$, where ${𝐴}^{\ast }$ is the conjugate transpose): same theorem, with $𝑄$ replaced by a unitary matrix $𝑈$ (${𝑈}^{\ast }𝑈=𝐼$).
• Normal matrices (${𝐴}^{\ast }𝐴=𝐴{𝐴}^{\ast }$): same theorem in ${ℂ}^{𝑛}$ — unitarily diagonalizable.

94.4. Why it matters

• Quadratic forms: ${\overrightarrow{𝑥}}^{𝑇}𝐴\overrightarrow{𝑥}$ simplifies to $\sum {𝜆}_{𝑖}{𝑦}_{𝑖}^2$ in the eigenbasis $𝑦={𝑄}^{𝑇}𝑥$. Definiteness is read off the signs of eigenvalues.
• PCA / dimensionality reduction: the covariance matrix is symmetric → orthogonal eigenbasis = principal components.
• Optimization: the Hessian is symmetric → its spectrum classifies critical points.
• Quantum mechanics: observables are Hermitian operators; their eigenvalues are the possible measured values.
• SVD extends the idea to non-square / non-symmetric matrices.

94.5. See also

• Symmetric Matrix
• Orthogonal Matrix
• Diagonalization
• Eigenvectors & Eigenvalues
• SVD