Let $G$ be an $n\times n$ complex matrix with block structure
$$G = \left( {\begin{array}{*{20}{c|c}} {\begin{array}{*{20}{l}} {i{r_1}}&{}&{}&{} \\ {}&{i{r_2}}&{}&{} \\ {}&{}& \ddots &{} \\ {}&{}&{}&{i{r_m}} \end{array}}& 0 \\ \hline 0 & {\begin{array}{*{20}{l}} {{T_1}}&{}&{}&{} \\ {}&{{T_2}}&{}&{} \\ {}&{}& \ddots &{} \\ {}&{}&{}&{{T_k}} \end{array}} \end{array}} \right)$$
with $r_1, r_2, \dots, r_m \in \mathbb{R}$ and $T_1, T_2, \dots, T_k$ being upper-triangular Jordan blocks of varying size for eigenvalues with strictly negative real part, e.g.
$${T_1} = \left( {\begin{array}{*{20}{c}} {{\lambda _1}}&1 \\ {}&{{\lambda _1}} \end{array}} \right), \;\Re\left(\lambda_1\right) < 0$$
Given a complex vector $\vec{v} \in \mathbb{C}^{n}$, the time evolution
$$\dot{\vec{v}}=G\vec{v}$$
can be rewritten as
$$\dot{\vec v} = \underbrace {iP^\dagger D{P}}_{{\Phi _1}}\vec v + \underbrace {P^{\perp\dagger} H{P^\perp}}_{{\Phi _2}}\vec v$$
with $P$ being the top $m$ rows of $\mathbb{I}_{n\times n}$, $P^\perp = \mathbb{I}_{n\times n} - P$, $D$ being $m\times m$ real diagonal, and $H$ being Hurwitz.
The linear operator $\Phi_1$ can therefore be thought of as a length-preserving evolution of $\vec{v}$ projected onto $P^\dagger P$, and $\Phi_2$ as a strictly length-decreasing evolution of $\vec{v}$ projected into the complementary space.
Obviously the number of elements in $\vec{v}$ that will not decay to zero as $t\rightarrow\infty$ is given by $\operatorname{rank} P$, which is $m$. We denote the number of elements in $\vec{v}$ that will decay to zero while evolving under $G$ as $z\left(G\right)$, where it is readily seen that $z\left(G\right) = n-m$.
Now suppose we instead consider the time evolution
$$\dot{\vec{v}}=\left( G + G'\right)\vec{v}$$
where $G'$ is skew Hermitian.
It is readily seen that $G'$ can be chosen so that $n - m \leq z\left(G + G'\right) \leq n$.
It is my conjecture that $\forall\; m,k \geq 1$ there does not exist any skew Hermitian $G'$ such that $z\left(G + G'\right) < n - m$, however I'm unsure of how to prove or disprove this.
Any assistance or insights would be much appreciated.