Suppose we have two n-by-n, symmetric, positive definite matrices (a.k.a. variance-covariance matrices). Are the elements always positively correlated?
For example, the following two has a correlation of 0.87.
\begin{bmatrix} 6.96461487 & -0.07257359 & -0.9801043 \\ -0.07257359 & 6.91825386 & -0.2198709 \\ -0.98010433 & -0.21987088 & 9.0856412 \\ \end{bmatrix}
\begin{bmatrix} 7.106961 & 1.7414630 & -2.7030161 \\ 1.741463 & 2.4653056 & -0.1420064 \\ -2.703016 & -0.1420064 & 6.6259570 \\ \end{bmatrix}
I simulated over 10,000 of such matrices and I still get positive correlation.
Is there a formal proof for that? Or is there a specific example where the matrices would be negatively correlated?
your basic variance - covariance matrix.
$\begin{bmatrix} \sigma_1^2 & cov_{1,2}\\cov_{1,2}&\sigma_2^2 \end{bmatrix}$
Which you could altertatively write as.
$\begin{bmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2\\\rho\sigma_1\sigma_2&\sigma_2^2 \end{bmatrix}$
If $\rho< 0$ it represents negative correlation.
$-1 \le \rho \le 1$ the matrix is positive definite.