Alternative parameterization for the covariance matrix via Euler angles

Question

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 $j = i, ..., p-1$.[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)

Answer 1

Those angles are sufficient to define the eigenvectors, to within normalization of the eigenvectors. Therefore you can use them to determine the eigenvectors, which after normalization, can be used to populate Q (there are two possibilities differing by a factor of -1 for the signs of elements for each column of Q, , but they cancel out due to multiplying Q and Q'). I presume the details are provided in the paper.