[Math] Calculating correlation matrix from covariance matrix – r>1

Question

I have been given a covariance matrix and asked to calculate the correlation matrix, but I get an error when doing this.

The covariance matrix is:

$$ \sum = \begin{bmatrix}4&6\\6&1\end{bmatrix}$$

and

$$Corr(X,Y)= \frac{Cov(X,Y)}{\sqrt{V(X).V(Y)}}$$

so here,

$$Corr(X,Y)= \frac{6}{\sqrt{4}}=3!!! $$

But, $$ -1 \le Corr(X,Y) \le 1 $$

So this is impossible. What am I missing here?

Answer 1

Your covariance matrix is not a covariance matrix.

It should be positive (semi) definite so both eigenvalues should be $\ge 0$, but $\det(\Sigma) < 0$ so the eigenvalues have opposite signs.