Under what conditions will the covariance matrix be identical to the correlation matrix?

Question

Under what conditions will the covariance matrix be identical to the correlation matrix?

I have been looking everywhere but no webpage or book seems to answer my question.

I just want to know when could this situation happen, and what that means for the variables.

Thanks

Answer 1

If all the variables $X_1 \ldots X_n$ are variance=1 - that is - they have the unit scale then $n\times n$ covariance matrix will be (in theory) identical to the correlation matrix. Note - they don't have be normally distributed for this to hold. That said, numerical differences due to the difference in algorithms ( i.e. formulas) may arise - so you definitely don't want to be doing $Cov == Corr$. The differences depend on how degenerate your system is and how much "divide-by-near-zero" error you accumulate.

Answer 2

The correlation between two random variabes $X,Y$ is $\operatorname{cor}(X,Y) = \dfrac{\operatorname{cov}(X,Y)}{\sqrt{\operatorname{var}(X)\operatorname{var}(Y)}}.$ Thus the correlation equals the covariance if the product of the variances is $1.$ But $\operatorname{cor}(X,X)= \dfrac{\operatorname{cov}(X,X)}{\sqrt{\operatorname{var}(X)\operatorname{var}(X)}} = \dfrac{\operatorname{var}(X)}{\operatorname{var}(X)} = 1,$ so this is equal to the covariance that you see in the numerator only if $\operatorname{var}(X) = 1.$ And similarly for $Y.$ Thus the $2\times 2$ matrix of correlations is equal to the $2\times 2$ matrix of covariances if, and only if, both of the variances are equal to $1.$

Since a correlation or a covariance is between only two random variables, the same conclusion holds for $n\times n$ in general: every entry in the $n\times n$ matrix is a covariance or a correlation of just two random variables.