# How to continuously rotate a bivariate normal distribution by changing its correlation matrix?

I want to create an animation in which a bivariate normal distribution rotate in xy plane by using matplotlib.

I don't want to alter the variances in the two principal components, all I want is to rotate the two principal components at the same pace.

How the correlation matrix should evolve in order to achieve this?

Suppose your initial covariance matrix is $$\Sigma$$.
Let $$R=\begin{bmatrix} \cos\theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$$.
Verify that $$R\Sigma R^T$$ rotates the principal component as $$\theta$$ changes. Try to prove this using eigenvalue decomposition of $$\Sigma$$.