# How does adding another row and column to a matrix affect the real part of its dominant eigenvalue?

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.
• @cluelessmathematician The matrix from the beginning of the answer is a matrix $B$, the matrix constituted by the left upper cell of $B$ is the matrix $A$. Next are presented eigenvalues of $B$ and $A$. Sep 23 '20 at 11:56