Can we find eigenvalues of $E_1+A$? $A$ is a real symmetric matrix and $E_1$ is a matrix whose $11$ entry is $1$ and rest are $0$. Further $A$ and $E_1$ don't commute. What can be said about eigenvalues of $A+E_1$
 A: 
What can be said about eigenvalues of $A+E_1$?

Not very much, I think. Just consider $A=\begin{bmatrix}0&1\\1&0\end{bmatrix}$. This matrix leaves alone the span of $\begin{bmatrix}1\\1\end{bmatrix}$, and negates the span of $\begin{bmatrix}-1\\1\end{bmatrix}$. So its eigenvalues are $1$ and $-1$.
But $A+E_1=\begin{bmatrix}1&1\\1&0\end{bmatrix}$. This matrix has characteristic polynomial $t^2-t-1$, with eigenvalues the golden ratios $\frac{1+\sqrt{5}}{2}$ and $\frac{1-\sqrt{5}}{2}$. It's kind of hard to imagine there being some systemic way in which these eigenvalues relate to the $-1$ and $1$ eigenvalues of $A$.

Note, if $A$ and $E_1$ do commute, then since $AE_1$ is a matrix of all zeros except the first column is $A$'s first column, and $E_1A$ is a matrix of all zeros except the first row is $A$'s first row, then you can deduce $A=\begin{bmatrix}a_{11}&0\\0&B\end{bmatrix}$. So $A$ has $a_{11}$ as an eigenvalue, and more eigenvalues that it shares with $B$. Then $A+E_1$ has these same eigenvalues from $B$, but replaces $a_{11}$ with $a_{11}+1$.
A: The matrix $A+E_{1,1}$ is still real symmetric. It can in fact be any real symmetric matrix that has at least one nonzero off-diagonal entry in its first column (because of the non-commutation condition). What can be said about its eigenvalues and eigenvectors is precisely what can be said about them for an arbitrary real symmetric matrix (real eigenvalues, orthogonal eigenspaces), plus the fact that the first standard basis vector is not an eigenvector.
