Consider some $n\times n$ matrix $\mathbf{A}$; denote its dominant (a.k.a. leading) eigenvalue $\lambda_{A,d}$. Consider another matrix $${B} = \left[ \begin{array}[cc] \ \mathbf{A} & \vec{c} \\ \vec{r}^{\top} & k \end{array}\right]$$
where $\vec{c}$ and $\vec{r} $ are column vectors from $\mathbb{R}^N$ and $k\in\mathbb{R}$. Denote the dominant eigenvalue of B by $\lambda_{B,d}$.
Does it then follow that $\text{Re}(\lambda_{A,d})\leq \text{Re}(\lambda_{B,d})$? I ran some simulations that seem to suggest this is true, but I am having trouble proving this.
Note: I've been considering the case where $\text{Re}(\lambda_{A,d})>0$. I don't think it matters, but if it does and/or makes your life easier, then please do feel free to impose that.
Thanks so much for taking the time to at least read this!
$$ \begin{pmatrix} 1 & -1\\ 2 & -1 \end{pmatrix} $$ The eigenvalues are $i$ and $-i$, whose real parts are zero, that is less than 1.
Notice: what you said is true for positive definite hermitian matrices