Suppose I know the eigenvalues and eigenvectors of an N-dimensional matrix M exactly. Now, I add a perturbation $\delta$ to just the (N, N) element of the matrix, M. In other words, the perturbation matrix is diag $(0,0,...\delta)$. The perturbation $\delta$ is real, positive and $\mathcal{O}(1)$.
Intuitively, I would expect that the perturbation's effect on the eigenvalues would decrease as the size of the matrix increases, as it affects only one element of a large matrix (even if $\delta$ is $\mathcal{O}(1)$). Some quick numerical plots for sample matrices suggest this as well. Is there a way to set up a formal expansion of the eigenvalues in $1/N$ to show this?
If it helps, the problem I am working on involves M being a real symmetric tridiagonal matrix, with most of the elements in each diagonal being equal (except for the (1, 1) and (N, N) elements). I have tried some of the common techniques for tackling tridiagonal Toeplitz matrices on this, but I have not been able to diagonalise it because of the end elements being different. I have a complete solution for the case in which $a = c + d$, but with arbitrary end elements $c$ and $d$, it seems to be insoluble.
$$M = \begin{pmatrix} c & b & 0 & 0 & \cdots \\ b & a & b & 0 &\cdots \\ 0 & b & a & b &\cdots \\ \vdots & & \ddots & \ddots & \ddots \\ 0 & & & b & a & b \\ 0 & & \cdots & & b & d \end{pmatrix}$$