Gradient and hessian of $\log(x^TAx)$

Question

I am working on a optimization problem which involves the gradient and hessian of $\log(x^TAx)$, where $x$ is an unknown vector and $A$ is a positive definite matrix. How can I derive them? Thanks!

Answer 1

First, let's define a quadratic form and calculate its differential. $$\eqalign{ \alpha &= x^TAx \cr d\alpha &= 2(Ax)^T dx \cr }$$ The objective function is the log of the preceeding, so its differential and gradient are easy to calculate. $$\eqalign{ \lambda &= \log\alpha \cr d\lambda &= \alpha^{-1}d\alpha \cr &= 2\alpha^{-1}(Ax)^T dx \cr g = \frac{\partial\lambda}{\partial x} &= 2\alpha^{-1}Ax \cr }$$ Now let's calculate the differential of $g$ and thence the hessian. $$\eqalign{ dg &= 2\alpha^{-1}A\,dx - 2Ax\,\,\alpha^{-2}\,d\alpha \cr &= 2\alpha^{-1}A\,dx - 4\alpha^{-2}Ax(Ax)^T\,dx \cr H = \frac{\partial g}{\partial x} &= 2\alpha^{-1}A - 4\alpha^{-2}Ax(Ax)^T \cr &= 2\alpha^{-1}A - gg^T \cr }$$