2
$\begingroup$

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 $|\mathbf{J}_{\mathbf{X}\to\mathbf{Y}}|$.

$\endgroup$

1 Answer 1

2
$\begingroup$

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 $Y=X^{\rm T}X$,

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

$\endgroup$
2
  • $\begingroup$ 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, 2014 at 19:06
  • $\begingroup$ that Jacobian vanishes $\endgroup$ Jun 4, 2014 at 19:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.