# Jacobian for a matrix transformation: Example of Cholesky decomposition

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!

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.