I would like to generally understand how the Jacobian of a matrix transformation can be computed. As a concrete example, consider the Transformation from a (correlation) matrix to its Cholesky factor: $\Omega \rightarrow L$, where $\Omega = LL^T$

The absolute determinant of the Jacobian is allegedly $\prod_{i=2}^K L_{ii}^{i-K}$, which tells me that the Jacobian must be triangular/diagonal.

I tried to vectorize $\Omega$ and $L$ in row-major order and compute the square matrix of partial derivatives with $\binom{K}{2}+ K$ columns and rows, but I ended up with something far from triangular.

Any help is appreciated!


1 Answer 1


Theorem: Let $X$ be a $p \times p$ symmetric positive definite matrix of functionally independent real variables and $T = (t_{ij})$ a real lower triangular matrix with $t_{jj}>0, j=1,...,p,$ and $t_{ij}, i \geq j$ functionally independent. Then

$$X=TT' \Rightarrow dX = 2^p \prod_{j=1}^p t_{jj}^{p+1-j} dT$$

There are two ways to get the above. I only outline the process here. (My personal habit is that I would like to provide a full and complete proof. But here you have to understand a lot of other transformations first, such as what is $Y=AX$. Otherwise you won't understand this question.)

Way 1: By considering the matrix of differentials we have

$$X=TT' \Rightarrow (dX) = (dT)T' + T(dT')$$

Now treat this as a linear transformation in the differentials, that is, $dX$ and $dT$ as variables and T a constant.

In this way, you need to know Jacobians of this transformation: $Y=AX'+XA'$, where $A=T, X=dT$.

Way 2: Again we start from $(dX) = (dT)T' + T(dT')$. Then $T^{-1} dX (T^{-1})' = T^{-1} (dT)+(dT')(T^{-1})'$.

Let $Z=U+U'$, where $Z=T^{-1} dX (T^{-1})', U=T^{-1} (dT)$.

Thus $J(X \rightarrow T)=J(dX \rightarrow dT)=J(dX \rightarrow Z)J(Z \rightarrow U)J(U \rightarrow dT) = \frac{1}{J(Z \rightarrow dX)} J(Z \rightarrow U)J(U \rightarrow dT)$.

Then you have to figure out what these three pieces are.

Like I said, I omit the complete proof here. But I can provide a reference for anyone who wants to know the whole story.

Reference: Mathai, A. M. (1997). Jacobians of matrix transformation and functions of matrix arguments. World Scientific Publishing Company.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .