Suppose $A \in \mathbb{R}^{n \times n}$ is an arbitrary matrix (not necessarily a square matrix), I want to solve the following optimization problem: $$ \min_{x \in \mathbb{R}^{n}} \frac{\langle x,Ax \rangle}{\| x \|_2 \| Ax \|_2}, $$ namely, I want to find a direction $x$ so that the matrix $A$ could rotate the direction to the maximal angle.
A similar question is asked in Maximum angle between a vector $x$ and its linear transformation $A x$, in which, however, the matrix $A$ is symmetric. In this special case, the maximal angle depends on the condition number of $A$. I also wonder whether this conclusion holds in the general case that $A$ is non-symmetric.
