# The Structure of the Jacobian Matrix in One-to-One Transformations

I currently study Searle's et al. (1992) book "variance components". In appendix S.d (page 474) they define Jacobian matrix of the transformation $$\Theta \rightarrow\Delta$$ as

$$J_{\Theta \rightarrow\Delta}=\left[_m \frac {\partial \Theta_i}{\partial \Delta_j}\right]_{i,j}$$

$$m$$ indicates that it is a matrix with the corresponding partial derivatives at $$(i,j)$$. Hence, $$J$$ is the Jacobian matrix of a function which takes as input the vector $$\Theta$$ and produces as output the vector $$\Delta$$. Or as the authors put it: parameters in $$\Theta$$ "are transformed in a one-to-one manner to the vector $$\Delta$$"

However, if I remind myself about the structure of a Jacobian matrix, amongst others here, then I find the following:

$$J=[\frac {\partial f}{\partial x_i}...\frac {\partial f}{\partial x_n}]$$

Here, J is the Jacobian matrix $$[m,n]$$ of $$f$$, where $$f:ℝ^n → ℝ^m$$ is a function which takes as input the vector $$x$$ and produces as output the vector $$f(x)$$.

Hence, I find the two definitions of the Jacobian matrix conflicting since

$$J_{\Theta \rightarrow\Delta} = J^{-1}$$

I do not believe that there is a mistake in Searle's et al. but cannot reconcile both definitions of a Jacobian matrix. What do I miss?

## 1 Answer

That's just because of how the Jacobian is used in probability. If $$\Delta = f(\Theta)$$ and $$p_\Delta, p_\Theta$$ are the two densities then

$$\mathbf{P}(\Delta \in A) = \int_A p_{\Delta}(\delta) \,d\delta = \mathbf{P}(\Theta \in f^{-1}(A)) = \int_{f^{-1}(A)} p_\Theta(\theta) \,d\theta$$

The change of variables theorem tells us that

$$\int_{f^{-1}(A)} p_\Theta(\theta) \,d\theta = \int_A p_\Theta(f^{-1}(\delta))\left| \det\left[ \frac{\partial \Theta_i}{\partial \Delta_j} \right] \right| \,d\delta.$$

So

$$p_\Delta(\delta) = p_\Theta(f^{-1}(\delta))\left| \det\left[ \frac{\partial \Theta_i}{\partial \Delta_j} \right] \right|.$$

Which means we don't want the Jacobian of $$f$$ as defined on Wikipedia, we want the Jacobian of $$f^{-1}$$.

• thanks for the answer. I was just wondering if you could recommend and literature (e.g. textbook) that would allow me to study this a bit more in detail. Cheers – DomB Jan 25 at 18:47
• @Dom Any multivariable calculus or analysis book covers the change of variables formula. You can also look at intro differential geometry books (e.g. Spivak's Calculus on Manifolds or Tu's An Introduction to Manifolds). – Trevor Gunn Jan 25 at 22:11