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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?

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1 Answer 1

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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 $\Sigma$.

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