Let $A$ be a matrix with a nonzero entry $t_0$ in the (1,1) entry and padded with zeros everywhere else; and let $B$ be a diagonalizable matrix, such that $VBV^T$ is diagonal. Is there anything we can say about the eigenvalues of $A+B$ if we know the eigenvalues of $A$ and $B$?
What if $A$ is merely a diagonal matrix?
Best Answer
In general not much. Consider for instance the matrix $$ B = \begin{bmatrix}0&a\\a&0\end{bmatrix}, $$ with characteristic polynomial $x^{2} - a^{2}$, so over the complex numbers, say, the eigenvalues are $\pm a$. (We have to take $a \ne 0$ to have it diagonalizable.) Add your $A$, you get $$ A + B = \begin{bmatrix}t_{0}&a\\a&0\end{bmatrix}, $$ with characteristic polynomial $x^{2} - t_{0} x - a^{2}$, which as $t_{0}$ ranges over the complex numbers, has as roots all pairs of complex numbers $u,v$ such that $u v = a^{2}$.
Even worse examples can be constructed, will possibly post later.