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 == Spectral Theorem ==   == Spectral Theorem == 
 Spectral Theorem is also sometimes called Principal Axis Theorem   Spectral Theorem is also sometimes called Principal Axis Theorem 
−  * In [[Linear Algebra]] a Spectrum is a set of [[Eigenvalues and EigenvectorsEigenvectors]] of a matrix  +  * In [[Linear Algebra]] a Spectrum is a set of [[Eigenvalues and EigenvectorsEigenvalues]] of a matrix 
   
   
Latest revision as of 23:05, 10 August 2017
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 PositiveDefinite
Sources