# Alternative parameterization for the covariance matrix via Euler angles

Using spectral decomposition, we can write any symmetric matrix as

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

where $Q$ is orthonormal, and

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

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

An alternative parameterization 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 $j = i, ..., p-1$.

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 $\Sigma$

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

• yes. It is a angles. edited Jul 8 '15 at 13:59
• You have to specify a convention for what those angles mean. Even in $\mathbb{R}^3$ there are multiple conventions. If you want to use the convention in the paper you reference, then you will have to explain it here in your post.
– whuber
Jul 8 '15 at 17:57