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.