How to annualize the correlation matrix?

If asset returns are daily, and the asset return covariance matrix, $$\Sigma$$, is annualized by $$\Sigma \times 252$$, do I also multiply the correlation matrix by 252 to annualize it?

No, because correlation is a unitless quantity. As you use volatilities to do the scaling, the $$\sqrt{252}$$ factor should already be taken into account in them.
If you take a correlation of 1 between two assets, multiplying your correlation matrix by a factor $$C \neq 1$$ risks either to underestimate correlations (by hiding perfect (anti)correlations) or have your matrix not making any sense (correlation greater than 1).
• To make it easy to understand, recall that correlation is $\frac{cov(x,y)}{\sqrt{var(x) var(y)}}$. So, the $252$ in the numerator and denominator will simplify. Sep 15 '20 at 9:07
• @develarist, Some important things to keep in mind: (1) correlations are between $-1$ and $1$, if you multiply by $\sqrt{252}$ it won't be a correlation anymore. (2) the rescaling by this $\sqrt{252}$ doesn't work if there is autocorrelation or mean-reversion, works only when iid. (3) To answer your question why it should be applied to the denominator as well, if the increments are iid, then it applies to the covariance but also the variance (particular case as $var(x) = cov(x, x)$). Sep 16 '20 at 8:28