# In PCA, why do we assume that the covariance matrix is always diagonalizable?

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?

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