Given a $n \times n$ matrix $A$ with eigenvalues $\lambda_k$ for $k = 1, 2, \dots, n$ we know the relationship between $\{\lambda_k\}$ and the eigenvalues of $A + t I$ where $I$ is the identity matrix and $t$ a scalar. I am wondering what happens when we replace $I$ with the exchange matrix $E$. That is, can we say anything about the eigenvalues of $A + t E$ in relation to the eigenvalues of $A$?
Context:
I am trying to prove (not sure that assertion is true or not) that singular Hankel matrices can be made invertible by perturbing the values of its antidiagonal entries (just like a singular $A$ can be made invertible by a perturbation $A + tI$).
$$\begin{bmatrix} f(2) & f(3) & f(4) &\dots &f(n+1) \\ f(3) & f(4) & \dots & & f(n+2) \\ \vdots & & & & \vdots \\ f(n+1) & f(n+2) & \dots & & f(2n) \end{bmatrix}$$
So within this context, $A$ has more structure than as stated above.