How do you translate a density from Cholesky factor to density of the matrix?

Question

Suppose $L$ is a random $p\times p$ lower triangular matrix, with known density, $f(L)$. To compute the density of $C=L L^{\top}$, one needs to use the change of density formula. This is a little bit hairy because $C$ and $L$ are matrices.

Digging around on the web, I found the formula $$ \frac{\mathrm{d}\operatorname{vech}\left(C\right)}{\mathrm{d}\operatorname{vech}\left(L\right)} = S_p \left[ \left(I_p \otimes L\right) T_{p,p} + L\otimes I_p \right] S_p^{\top}, $$ where $S_p$ is the 'elimination matrix' which takes $\operatorname{vec}$ to $\operatorname{vech}$, and $T_{p,p}$ is the 'transpose matrix': $T_{m,n}\operatorname{vec}\left(X\right) = \operatorname{vec}\left(X^{\top}\right),$ for $m\times n$ matrix $X$.

Computing the determinant of this ugly product is beyond my capabilities at the moment. I assume this is a problem with a known solution, however.

Answer 1

If $L$ is a $p\times p$ lower triangular matrix and $C=LL^T$, then the Jacobian is given in Equation (55) of Henderson and Searle (1978) (see page 16) as $$J(C\rightarrow L) = 2^p \prod_{i=1}^p {l\,}_{i,i}^{p-i+1}.$$ They note that this result was given as Theorem 4 of Deemer and Olkin (1951).

This result is also shown in other places, such as Theorem 4.2 of Njoroge (1988), who explicitly stipulates that $C$ be positive definite symmetric and that $L$ have positive diagonal elements.

References

W.L. Deemer and I. Olkin (1951) The Jacobians of certain matrix transformations useful in multivariate analysis. Based on lectures of P.L. Hsu at the University of North Carolina, 1947, Biometrika 38:345--367.

H.V. Henderson and S.R. Searle (1978) Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics. Technical Report BU-643-M. Biometrics Unit, Cornell University.

The published version of the technical report seems to be:

H.V. Henderson and S.R. Searle (1979) Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics. The Canadian Journal of Statistics 7(1):65--81.

M.N. Njoroge (1988) On Jacobians connected with matrix variate random variables. Thesis.