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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?

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    what can you say about the eigenvalues of $\lambda A + (1-\lambda)B$ if $A$ and $B$ are Hurwitz? – 2017-01-30
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    @LinAlg, Thanks for your comment. Here you have just apply the definition of convex set. If this is proved, then we have the result. But how to prove this? Do you have any hint ? – 2017-01-30
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    The eigenvalues of $\lambda A$ are trivial to find, so it boils down to eigenvalues of sums of matrices, right? – 2017-01-30

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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 $t<0$.

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    Thank you! The condition is false for $t = \frac{1}{3}$ for example. – 2017-01-30
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    @Igor you mean $t=-1/3$ – 2017-01-30
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    Yeah! That's right! – 2017-01-30