Let me add a little more detail:
I have an arbitrarily sized nxn matrix. This matrix is symmetric and binary (only 0's and 1's). The eigenvalues and eigenvectors are known. An element is changed in this matrix. For example, a 0 turns to 1 or a 1 turns to 0. If an element off the main diagonal is changed, its symmetric counterpart changes as well. If the element is on the diagonal only 1 element is changed (bc it doesn't have a symmetric counterpart).
Now, what is a function that outputs the new eigenvalues/eigenvectors after the change is made? Not necessarily, but preferably, a multivariable function of the old eigenvalues and the change that was made.
I've come up with solutions to specific examples (ie, where the matrix is 3x3), but deriving a general solution is more difficult.