= Correlation matrix
{title2=$C=D^{-1/2}\Sigma D^{-1/2}$}
For a random vector with <covariance> $\Sigma$ and positive coordinate <variances>, let $D=\operatorname{diag}(\Sigma)$. Its <correlation matrix> is the <covariance> of $D^{-1/2}(X-\mathbb EX)$, so it is positive semidefinite with unit diagonal. Each entry is the <correlation> of the corresponding pair of variables. In particular $\operatorname{tr}(C)=p$, explaining why an <eigenvalue> in correlation-based <principal component analysis> represents that <eigenvalue> divided by $p$ of the total <variance>. The empirical version is the <sample correlation matrix>.
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