Motivation: I am working on a research problem and have been stuck for a while. I hope someone can help, as it requires only linear algebra. :)


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 always stable and $(I+\alpha H\_d)$ is always positive definite. Ideally I would have liked to use Sylvester's Law of Inertia to conclude, as suggested [here](https://mathoverflow.net/questions/181125/eigenvalues-of-product-of-diagonal-positive-matrix-and-symmetric-matrix), 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.



Stability of a matrix product
It's true for all $\alpha \ge 0$. $G$ has the same eigenvalues as $(1+\alpha H\_d)^{1/2} H (1+\alpha H\_d)^{1/2}$, which is positive semidefinite.


