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In Principal Component Analysis, what is the justification for the assumption that the covariance matrix is always a diagonalizable matrix?

What happens when the covariance matrix is not diagonalizable, i.e. its eigenvectors do not span the complete vector space?

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Covariance matrix is a symmetric matrix, hence it is always diagonalizable.

In fact, in the diagonalization, $C=PDP^{-1}$, we know that we can choose $P$ to be an orthogonal matrix.

It belongs to a larger class of matrix known as Hermitian matrix that guarantees that they can be diagonalized.

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