Smoothness of the function

Question

Suppose $f(x)=x^TAx$, where $A = \begin{bmatrix}100& 0& ...0\\ 0&1 &...0\\ ...\\ 0&0&...1\end{bmatrix}$ is an identity matrix except $A_{11}=100$ instead of 1. How can I compute the smoothness of this function?

Answer 1

$A$ is the Hessian of $f$, and it is positive definite, so it is a strictly convex function for all $x$. It is everywhere differentiable, with derivative $\nabla f(x) = 2A'x$. The eigenvalues are on the diagonal, with the largest being 100 and the smallest being 1, so the condition number is 100/1, so I would be aware of the numerical accuracy of matrix calculations if you aren't using closed form solutions.

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