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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.

Matthias Mitterbacher
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1 Answers1

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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 $$\eqalign{ X &= CC^T \cr \cr dX &= dC\,C^T + C\,dC^T \cr \cr {\rm vec}(dX) &= (C\otimes I)\,{\rm vec}(dC) + (I\otimes C)\,{\rm vec}(dC^T) \cr &= \Big[(C\otimes I) + (I\otimes C)K_k \Big]\,{\rm vec}(dC) \cr \cr L_k{\rm vec}(dX) &= L_k\Big[(C\otimes I) + (I\otimes C)K_k \Big]\,D_k{\rm vech}(dC) \cr {\rm vech}(dX) &= L_k\Big[(C\otimes I) + (I\otimes C)K_k \Big]\,D_k\,{\rm vech}(dC) \cr \cr \frac{\partial{\rm vech}(X)}{\partial{\rm vech}(C)} &= L_k\Big[(C\otimes I) + (I\otimes C)K_k \Big]\,D_k \cr }$$

lynn
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  • Thank you for the answer. I looked up the definitions & properties of the elimination, commutation and duplication matrices and checked all the steps you made. I get the same result as you. Thanks again! – Matthias Mitterbacher Feb 24 '17 at 11:48
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    It should be noted that the $D_k$ matrix used above is not the standard Duplication matrix, but rather a Triangular Duplication matrix $$D_\Delta = {\rm Diag}(D_kL_k)\cdot D_k$$ The difference is that $D_k$ generates the elements of a symmetric matrix from ${\rm vech}(C),$ while $D_\Delta$ generates those of a triangular matrix. – greg Mar 19 '21 at 19:46