How to efficiently use inverse and determinant in Eigen?

Question

In Eigen there are recommendations that warn against the explicit calculation of determinants and inverse matrices.

I'm implementing the posterior predictive for the multivariate normal with a normal-inverse-wishart prior distribution. This can be expressed as a multivariate t-distribution.

In the multivariate t-distribution you will find a term |Sigma|^{-1/2} as well as (x-mu)^T Sigma^{-1} (x-mu).

I'm quite ignorant with respect to Eigen. I can imagine that for a positive semidefinite matrix (it is a covariance matrix) I can use the LLT solver.

There are however no .determinant() and .inverse() methods defined on the solver itself. Do I have to use the .matrixL() function and inverse the elements on the diagonal myself for the inverse, as well as calculate the product to get the determinant? I think I'm missing something.

Answer 1

If you have the Cholesky factorization of Sigma=LL^T and want (x-mu)^T*Sigma^{-1}*(x-mu), you can compute: (llt.matrixL().solve(x-mu)).squaredNorm() (assuming x and mu are vectors).

For the square root of the determinant, just calculate llt.matrixL().determinant() (calculating the determinant of a triangular matrix is just the product of its diagonal elements).