Efficient Computation of Square Root of Covariance Matrix

[foges]

So I need to calculate the square root of the covariance matrix $\sqrt{\Sigma_tR\Sigma_t}$ (the matrix square root, not the element-wise square root). $\Sigma_t$ is a diagonal matrix with the square root of the variance on the diagonal (these values are time dependent) and $R$ is the correlation between my variables (this is assumed to be independent of time). Here is an example:

$\sqrt{\left(\begin{array}{cc}\sigma_1 & 0 \\ 0 & \sigma_2\end{array}\right) \cdot \left(\begin{array}{cc}1 & \rho \\ \rho & 1 \end{array}\right) \cdot \left(\begin{array}{cc}\sigma_1 & 0 \\ 0 & \sigma_2\end{array}\right) }$

Now the thing is, it is awfully slow to recalculate the square root of this matrix for every time step. Seeing as my correlation is constant I was thinking there might be a more computationally efficient method of calculating this root, but haven't been able to come up with anything. Does anyone have any suggestions?

 

[aesir]

If the problem is 2 dimensional, then you can diagonalize the matrix and get an analytical result: http://www.wolframalpha.com/input/?i=diagonalize+{{a%2Cb}%2C{b%2Cc}}

 

[Stephen Tashi]

Another formulation of that is http://en.wikipedia.org/wiki/Square_root_of_a_2_by_2_matrix, if we may trust the Wikipedia. I haven't tried to check that article.