I have a convex polytope defined by $\mathbf{Ax \leq b}$ (row-wise)

The log-barrier function is defined as:

$$\phi(x) =-\sum{\log(b_i - a_ix)}$$

The Hessian of the log-barrier is :

$$\nabla^2\phi(x) = \text{...magic algebra} = A^TD^2A$$

where D = diag($\frac{1}{b_i-a_ix}$)

I understand that the log-barrier is sort of a measurement of how extreme the point $x$ is from the boundary of the polytope (as $\phi$ tends to $\infty$ as $x$ tends to the boundary). However, my question is:

How do I intuitively understand this Hessian?

More background: I am trying to understand intuitively what the Dikin Ellipsoid is (https://www.cs.cmu.edu/~ggordon/10725-F12/scribes/10725_Lecture23.pdf, http://stanford.edu/class/ee364b/lectures/dikin_slides.pdf )


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