Derivative of Symmetric Positive Definite Matrix w.r.t. to its Lower Triangular Cholesky Factor

Setup:

Let $k\in{}\mathbb{N}$ be a natural number, and let $\mathrm{M}_{k,k}(\mathbb{R})$ denote the set of $k\times{}k$ matrices over the field of real numbers.

Let $X\in{}\mathrm{M}_{k,k}(\mathbb{R})$ be a symmetric, positive definite matrix.

Let $L\in{}\mathrm{M}_{k,k}(\mathbb{R})$ be the lower triangular Cholesky factor of $X$, such that $X = LL'.$

Question:

What is the analytical expression for the $[k(k+1)/2] \times{} [k(k+1)/2]$ Jacobian matrix

$$\frac{\partial{}\mathrm{vech}(LL')}{\partial{}\mathrm{vech}(L)'}$$

where $\mathrm{vech}(\cdot{})$ is the half-vectorization operator that stacks the lower triangular part of its square argument matrix.

Solution 1:

You should be familiar with the Duplication ($D_k$), Elimination ($L_k$), and Commutation ($K_k$) matrices used to manipulate Kronecker-vec/vech expressions.

Since the symbol $L$ will be used for Elimination matrices, I'll be using $X=CC^T$ as the Choleski factorization.

Start by taking the differential of the matrix relation, then apply vectorization $$\eqalign{ X &= CC^T \cr \cr dX &= dC\,C^T + C\,dC^T \cr \cr {\rm vec}(dX) &= (C\otimes I)\,{\rm vec}(dC) + (I\otimes C)\,{\rm vec}(dC^T) \cr &= \Big[(C\otimes I) + (I\otimes C)K_k \Big]\,{\rm vec}(dC) \cr \cr L_k{\rm vec}(dX) &= L_k\Big[(C\otimes I) + (I\otimes C)K_k \Big]\,D_k{\rm vech}(dC) \cr {\rm vech}(dX) &= L_k\Big[(C\otimes I) + (I\otimes C)K_k \Big]\,D_k\,{\rm vech}(dC) \cr \cr \frac{\partial{\rm vech}(X)}{\partial{\rm vech}(C)} &= L_k\Big[(C\otimes I) + (I\otimes C)K_k \Big]\,D_k \cr }$$