Gaussian Distribution Under Orthogonal Transformation

Question

Let $\mathbf{H}\in \mathbb{R}^{n\times n}$ be a random matrix whose every element has a Gaussian distribution with mean $m_{ij}$ and variance $\sigma^2$ i.e. $h_{ij}\sim\mathcal{N}\left(m_{ij},\sigma^2\right)$, where $h_{ij}$ is the $(i,j)^{th}$ element of $\mathbf{H}$. Let us denote the probability density function(pdf) of $\mathbf{H}$ as $p_{\mathbf{H}}$. Now define a random matrix as \begin{equation} \mathbf{Y} = \mathbf{U}^T\mathbf{HU} \end{equation} where, $\mathbf{U}\in\mathbb{R}^{n\times m}, n > m$ is an orthogonal matrix. How does the pdf of $\mathbf{Y}$, denoted as $p_{\mathbf{Y}}$, relate to $p_{\mathbf{H}}$. I know that $p_{\mathbf{Y}} =p_{\mathbf{H}}$ when $m=n$ as per theorem 2.1.6, page 58, Aspects of Multivariate Statistical Theory, Robb J. Muirhead. But can we say the same when $n>m$?