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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!

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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.

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