# derivative of quadratic form with regard to inverse of lower-triangular matrix

I have a quadratic form of the form $Q(\Sigma; x, \mu) = (x-\mu)'\Sigma^{-1}(x-\mu)$ where $\Sigma$ is a positive-definite non-singular matrix with (modified) Cholesky decomposition $\Sigma = LDL'$ with $L$ being a lower-triangular matrix (with diagonals all unity).

I want to find the derivative of the above quadratic form with respect to the entries in $L$. That is, I want to find

$$\frac\partial{\partial L} Q(\Sigma;x,\mu).$$

Is this an established result? are there references I could look at?

Many thanks!

• you might find the answer here math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf Jul 27, 2015 at 15:21
• I have looked at this book. The derivatives are only wrt unconstrained matrices. Do the constraints (of lower triangularity of the matrix) not matter? Jul 27, 2015 at 15:41
• I guess I am a bit confused also -- the vech operator (defined for the lower-triangle) should apply. But that is typically defined for symmetric matrices. Does this matter that this is not a symmetric matrix but we are only interested in one triangle (and the other triangle is a constant zero). Jul 27, 2015 at 15:53
• I would suggest you should try out with some 2X2 matrices first and from there prove the formula --that how we learn maths :) Jul 27, 2015 at 16:15
• So, I guess try and reinvent the wheel is what you would suggest as the best option. I was planning to do that, but i wanted to know if there were standard results already available. Jul 27, 2015 at 16:46