Let : $$f(\mathrm{x}) = \frac{1}{4}\|\mathrm{A - xx^T\|_F,A\succ 0,x\in R^n}$$ where $\| \cdot \|_F$ is the Frobenius norm. Please calculate all critical points that satisfy $\nabla f(\mathrm{x}) = 0$, and determine if they are optimal points.
Now I have obtained the following equation: $$\nabla f(\mathrm{x}) = \mathrm{(xx^T-A)x}$$ $$\nabla^2f(\mathrm{x}) = 2\mathrm{x}\mathrm{x}^T+\mathrm{x}^T\mathrm{x}\cdot I-A$$
I already know that $\nabla f(\mathrm{x}) = 0$ means $\mathrm{x} = 0$ or $\mathrm{x}$ is an eigenvector of A.
But I can't tell if the Hessian matrix is positive definite after these eigenvectors are substituted into the Hessian matrix.