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Using spectral decomposition, we can write any symmetric matrix as

$$\Sigma = Q \Lambda Q^{\top},$$

where $Q$ is orthonormal, and

$$\Lambda = \operatorname{diag}(\lambda_1, ..., \lambda_p)$$

with $\lambda_1 \geq ... \geq \lambda_p \geq 0$.

An alternative parametrization can be made for the covariance matrix in terms of eigenvalues $\lambda_1,...,\lambda_p$ and $Q$ can be expressed using Euler angles in terms of $p(p-1)/2$ angles $\theta_{ij}$, where $i = 1,2,...,p-1$ and $j = i, ..., p-1$. [1]

Can someone elaborate on this method such that, given a function with $p$ eigenvalues and $p(p-1)/2$ angles, I can build a valid $\Sigma$.

[1]: Hoffman, Raffenetti, Ruedenberg. "Generalization of Euler Angles to N‐Dimensional Orthogonal Matrices". J. Math. Phys. 13, 528 (1972).

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  • $\begingroup$ Do you have a link to a pdf of that paper? $\endgroup$ – Neal Lawton Jul 8 '15 at 6:57

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