Let $\mathbf{X}$ be an $m\times m$ random matrix full rank matrix, having the density function $f_{\mathbf{X}}(X)$. Also, let $\mathbf{W}$ be a deterministic $k\times m$ matrix of rank $k$ and $k<m$. What is the distribution of $\mathbf{Y}=\mathbf{W}\mathbf{X}\mathbf{W}^T$? Basically my question is how to calculate the Jacoubian $|J(X→Y)|$.

  • $\begingroup$ unlike in your previous question, --- mathoverflow.net/questions/169041/… --- you now have a linear transformation, so the Jacobian is constant and only affects the normalization of your distribution; so you don't really need it, but it's $$\left|\left|\frac{\partial WXW^{\rm T}}{\partial X}\right|\right|={\rm det}\,(WW^{\rm T})$$ $\endgroup$ – Carlo Beenakker Jun 12 '14 at 20:13
  • $\begingroup$ Thanks Carlo. I still can not write the distribution of $\mathbf{Y}$ since the transformation from $\mathbf{X}$ to $\mathbf{Y}$ is not invertible. What is $f_\mathbf{Y}(\mathbf{Y})$? $\endgroup$ – Peter Jun 12 '14 at 23:07
  • $\begingroup$ you'll have to integrate over the degrees of freedom of $X$ that do not appear in $Y$; without knowing $f_X(X)$ there's no way to tell you what the outcome of the integration will give. $\endgroup$ – Carlo Beenakker Jun 13 '14 at 0:48

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