Can something be said about the eigenvalues of a matrix of the form:
$$ C = \begin{pmatrix} A & \mathbb{1} \\ \mathbb{1} & B \end{pmatrix} $$ where all the blocks are square and of the same size, and the matrices $A,B$ are symmetric?
In particular, is there some relation between the eigenvalues of $C$ and those of $A$ and $B$ separately?
Here $\mathbb{1}$ denote identity matrices.
For context, I encountered a matrix of this form as the Hessian of a certain function for which I am trying to determine the conditions for a critical point to be a minimum. That's why I need to evaluate the eigenvalues of $C$.
Let the blocks be $n\times n$. We have: $$\left\vert\begin{pmatrix} A & I_{n} \\ I_{n} & B\end{pmatrix} - \lambda I_{2n}\right\vert = \begin{vmatrix} A - \lambda I_{n} & I_{n} \\ I_{n} & B - \lambda I_{n} \end{vmatrix}$$ For general block matrix $\begin{pmatrix}A & B \\ C & D\end{pmatrix}$ such that $AC=CA$, we have $\begin{vmatrix} A & B \\ C & D\end{vmatrix} = |AD - BC|$. The above matrix clearly satisfies this, so: $$\begin{vmatrix}A-\lambda I_{n} & I_{n} \\ I_{n} & B-\lambda I_{n}\end{vmatrix} = \vert (A - \lambda I_{n})(B -\lambda I_{n}) - I_{n}^{2}\vert =\vert AB - \lambda(A+B)+(\lambda^{2}-1)I_{n}\vert = 0$$ As expected, this is a degree $2n$ polynomial in $\lambda$, and the roots don't seem to have any special relationship to the eigenvalues of $A$ or $B$.