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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

Sum of Rank One Matrices

We can look differently at the results of Eigendecomposition of $A$

• $A = Q \Lambda Q^T = \begin{bmatrix} | & & | \\ \mathbf q_1 & \cdots & \mathbf q_n \\ | & & | \\ \end{bmatrix} \begin{bmatrix} \lambda_1 & & \\ & \ddots & \\ & & \lambda_n \\ \end{bmatrix} \begin{bmatrix} - & \mathbf q_1^T & - \\ & \vdots & \\ - & \mathbf q_n^T & - \\ \end{bmatrix}$
• 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

• 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