Let $H$ be a real, invertible and positive semi-definite matrix, in the sense that its symmetric part $S$ is positive semi-definite. Consider the matrix $$ G = (I+\alpha H_d)H $$ for some $\alpha > 0$, where $H_d$ is the diagonal part of $H$ (obtained by setting all off-diagonal entries of $H$ to $0$). Prove or disprove that $G$ is stable for small enough $\alpha$, in the sense that its eigenvalues have non-negative real part.

I can neither prove this nor find a counter-example. It is definitely true for $2 \times 2$ matrices, and I think also for $3 \times 3$. In general, there are two special cases worth mentioning:

  • If $H$ has no pure imaginary eigenvalues then its eigenvalues have positive real part, and so does $G$ for small enough $\alpha$. This holds in particular if $H$ is symmetric.
  • If $H$ is antisymmetric then $H_d = 0$ so $G = H$, which has pure imaginary eigenvalues, with zero real part as required.

Note that $H$ is stable and $(I+\alpha H_d)$ is positive definite. Ideally I would have liked to use Sylvester's Law of Inertia to conclude, as suggested here, but neither $H$ nor $G$ are symmetric. I also tried some other sufficient conditions for stability like diagonal dominance, but this does not hold in general.

  • $\begingroup$ Cross-posted to MO $\endgroup$ Jul 1 '18 at 17:28
  • $\begingroup$ I'm quite new here - is that not okay? Since it's both a research and general question, I thought people from both communities might find it interesting. $\endgroup$
    – smalldog
    Jul 1 '18 at 17:30
  • 2
    $\begingroup$ It's OK if you tell both communities that you've cross-posted, including a link to the posting on the other site. See e.g. this post on meta $\endgroup$ Jul 1 '18 at 21:48
  • $\begingroup$ @RobertIsrael ...and if you do it only after not receiving an answer for a few days on the first website. $\endgroup$ Jul 2 '18 at 7:05

Note that $G$ is similar to the matrix $$ (I + \alpha H_d)^{-1/2}G(I + \alpha H_d)^{1/2} = (I + \alpha H_d)^{1/2}H(I + \alpha H_d)^{1/2} $$ Because $H$ is positive semidefinite, any matrix of the form $MHM^T$ is positive semidefinite, and we can take $M = (I + \alpha H_d)^{1/2}$ in particular. Because $G$ is similar to a positive semidefinite matrix, it must be stable.

I see no justification that the matrix $G'$ should generally be positive stable. In fact, if $H$ fails to be invertible, then $G$ cannot be positive stable since it also fails to be invertible.

If $G$ is positive stable, then we can certainly choose $\alpha$ sufficiently small to make $G'$ positive stable since we have $\lim_{\alpha \to 0^+}G'(\alpha) = G$.

The trick we applied before doesn't work unless we happen to know that $H$ is symmetric. If $H$ isn't symmetric, then the matrix $(I + \alpha(H_d - H))$ also fails to be symmetric.

  • $\begingroup$ Thanks for your perfect reply. I edited my question to ask one further thing before I accept your answer! $\endgroup$
    – smalldog
    Jul 1 '18 at 17:06
  • $\begingroup$ @Nao I've added a bit addressing your edit. In the future, however: if you're going to add something to your original question, it would be best to create a separate post with that followup question (you may want to include a link to the original question). $\endgroup$ Jul 1 '18 at 17:53
  • $\begingroup$ Thanks for your comment, and I will avoid adding an edit next time. I will post a separate question now to see whether someone can find a counter-example where $G'$ is not positive stable... I have tried hard to find one. Note that $H$ is assumed invertible, by the way. $\endgroup$
    – smalldog
    Jul 1 '18 at 17:56
  • $\begingroup$ @Nao I missed that bit $\endgroup$ Jul 1 '18 at 17:57
  • $\begingroup$ @Nao have you tried generating random matrices $H$? $\endgroup$ Jul 1 '18 at 17:58

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