Spectral Theorem
Spectral Theorem is also sometimes called Principal Axis Theorem
Theorem:
- Every Symmetric Matrix can be factorized as $A = Q \Lambda Q^T$
- with real eigenvalues $\Lambda$ and orthonormal eigenvectors in the columns of $Q$
The factorization is Eigendecomposition
- Spectral Theorem is a special case for symmetric matrices
- See the proof in the Symmetric Matrices article
We can look differently at the results of Eigendecomposition of $A$
- [math]A = Q \Lambda Q^T = \begin{bmatrix}
| & & | \\
\mathbf q_1 & \cdots & \mathbf q_n \\
| & & | \\
\end{bmatrix}
\begin{bmatrix}
\lambda_1 & & \\
& \ddots & \\
& & \lambda_n \\
\end{bmatrix}
\begin{bmatrix}
- & \mathbf q_1^T & - \\
& \vdots & \\
- & \mathbf q_n^T & - \\
\end{bmatrix}[/math]
- can represent it as $A = Q \Lambda Q^T = \sum \lambda_i \mathbf q_i \mathbf q_i^T$ - sum of Outer Products
- each of these outer products can be seen as a Projection Matrix
- a projection matrix is $P_i = \cfrac{\mathbf q_i \mathbf q_i^T}{\| \mathbf q_i \|^2} = \mathbf q_i \mathbf q_i^T$
- so symmetric matrix can be represented as a combination of mutually orthogonal projection matrices
Applications
Principal Component Analysis
- The Spectral Theorem guarantees that we will find an orthogonal basis in PCA
- Because the Covariance Matrix $C = \cfrac{1}{n - 1} X^T X$ is symmetric and Positive-Definite
Sources