Determine whether the set of matrices $A\in \mathbb{R}^{n \times n}$ which are Hurwitz, denoted here by $\mathcal{H}$, (i.e., all their eigenvalues lies in the open half-plane $\mathbb{C}^-$) is convex.
I have no idea on how to start. Does anyone have an idea on how to approach this?
The answer will be no, so you should try to find an example. For instance, consider $$ A=\pmatrix{t&1\\0&t}, \quad B=\pmatrix{t&0\\1&t} $$ for an appropriate value of $t<0$.