Factorisation of Gaussian random matrix into random Hermitian and correction factor

Question

By the Bartlett decomposition, one has that for $k \leq n$ and $\mathbf{\Gamma}_{n\times k} \in \mathbb{R}^{n\times k}$ a standard Gaussian matrix with independent entries

$$\mathbf{\Gamma}_{n\times k} \sim \mathbf{Q}_{n\times k}\mathbf{R}_{k\times k}$$

for $\mathbf{Q}_{n\times k}, \: \mathbf{R}_{k\times k}$ statistically independent, $\mathbf{Q}_{n\times k}$ uniformly distributed on the Stiefel manifold $\mathbb{V}_{k}^n$ and $\mathbf{R}_{k\times k}$ random upper-diagonal, such that the $(i,i)$-th diagonal entry is distributed as a $\chi^2_{n-i+1}$ random variable, whilst the $(i,j)$-th super-diagonal entry is a standard Gaussian and all entries are mutually independent.

Question: suppose $n = k$. Is there any known factorisation of a square Gaussian matrix into the product of a random Hermitian and a correction factor as below?

$$\mathbf{\Gamma}_{n\times n} \sim \mathbf{Q}_{n\times n}\mathbf{B}_{n\times n}\mathbf{Q}_{n\times n}^T\mathbf{\Delta}_{n\times n}$$

for $\mathbf{Q}_{n\times n}$ as before, an arbitrary but fixed diagonal $\mathbf{B}_{n\times n} \neq \text{Id}$ and random $\mathbf{\Delta}_{n\times n}$, independent of $\mathbf{Q}_{n\times n}$ (which may depend on $\mathbf{B}_{n\times n}$). Thanks in advance.

Answer 1

Write the SVD of $\Gamma$, say $\Gamma = \sum_i q_i s_i v_i^T$. with $s_1,...,s_n>0$ the singular values and $q_i, v_i$ are the left and right singular vectors. If $Q=[q_1|...|q_n]$, $B=diag(s_1,...,s_n)$ and $V=[v_1|...|v_n]$ then $\Gamma=QBV^T$. The crux of the matter is that $(Q,B,V)$ are mutually independent with $Q,V\in O(n)$.

From here, $\Gamma=QBQ^T QV^T$ to obtain the desired form. It remains to show that $\Delta=QV^T$ is independent of $Q$.