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In this question I want to investigate the linear subspace of the Hurwitz matrix family.

That is to say, suppose $M_H = \{A \in R^{n \times n}: \operatorname{Re} \lambda_i(A) \leq 0, \forall i \leq n\}$ is the hurwitz matrix family, and $U, V$ are two matrices satisfying \begin{equation} \forall \alpha \in R, U + \alpha V \in M_H \end{equation}

What can we say about the matrix $V$ here?

This problem should have some untrivial solutions, since I have already proved that if $U$ is negative-definite and $V$ is skew-symmetric, then $U,V$ will satisfy the condition.

Remark:

  1. What I considered here is only the real case. However it can also be helpful if you generalize it to general $M_n(C)$.

  2. A proof of the case where $V$ is skew-symmetric can be find here: Proving that the sum of a normal, Hurwitz stable matrix and a skew-Hermitian matrix is again Hurwitz?

EggTart
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  • Some sufficient conditions: 1 - both $U$ and $V$ are symmetric. 2 - both $U + U^T$ and $V + V^T$ are negative definite. 3 - $UV = VU$. – Ben Grossmann Aug 02 '23 at 22:57
  • I think this is not the case. What I ask here is that $U + \alpha V$ has eigenvalues with nonpositive real part for every $\alpha \in R$, so this $\alpha$ here can be both positive and negative. Thus $V+V^T$ can not be negative definite. – EggTart Aug 02 '23 at 23:21
  • You're right, my mistake – Ben Grossmann Aug 03 '23 at 16:53

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