1.4 Eigenvalues and eigenvectors
You can explain why PCA works without hand-waving.
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Eigenvectors are the directions a transformation only stretches, and for covariance matrices those are the directions of greatest variance — which is exactly what principal component analysis exploits. Positive definiteness is the property that keeps covariance matrices and many loss surfaces well behaved. It sits here because the spectral view unlocks PCA, PageRank and stability analysis at once. The confusion to avoid is memorising the definition without the geometry; the stretching picture is the entire point.
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Eigendecomposition and diagonalization
Some directions are only stretched by a transformation, never rotated, and finding them lets a matrix be rewritten as a stretch along those directions. This is the decomposition the rest of the topic uses.
Spectral theorem for symmetric matrices
Symmetric matrices always have a full set of perpendicular special directions and real stretch factors, which is a stronger guarantee than the general case. Covariance matrices are symmetric, which is why this matters here.
Positive definiteness and covariance matrices
A positive definite matrix stretches every direction rather than flipping any, and covariance matrices have this property. It is also the condition under which certain optimisation methods are guaranteed to work.
Power iteration and PageRank as an example
Repeatedly applying a matrix converges on its dominant direction, which is enough to build a ranking algorithm. It is a satisfying demonstration that this abstract machinery does concrete work.
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