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Let $\mathbf{X}$ be an $m\times k$ random matrix ($m>k$) of rank $k$, having the density function $f_\mathbf{X}(X)$. What is the distribution of $\mathbf{Y}=\mathbf{XX}^T$? Basically my question is how to calculate the Jacoubian $|\mathbf{J}_{\mathbf{X}\to\mathbf{Y}}|$.

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Jacobian, for $X$ an $m\times k$ real matrix with $m\geq k$

$$\left|\left|\frac{\partial X^{\rm T}X}{\partial X}\right|\right|=\left[{\rm det}\,(X^{\rm T}X)]\right]^{(1-m+k)/2}$$

see, for example,

A.M. Mathai, Jacobians of Matrix Transformations and Functions of Matrix Argument (World Scientific Publishing, 1997).

so if $P(X)dX=F(X^{\rm T}X)dX$ and $Y=X^{\rm T}X$,

$$P(Y)dY=F(Y)\left({\rm det}\,Y\right)^{(m-k-1)/2}dY$$

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  • $\begingroup$ Thanks for your response. But I want the answer for the rank deficient matrix $\mathbf{X}\mathbf{X}^T$, Not the full rank matrix $\mathbf{X}^T\mathbf{X}$. $\endgroup$ – Peter Jun 4 '14 at 19:06
  • $\begingroup$ that Jacobian vanishes $\endgroup$ – Carlo Beenakker Jun 4 '14 at 19:24

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