Response of eigenvalues under perturbation In this article about sensitivity of eigenvalues under perturbation, they represent the perturbation by replacing $A$ with $A + \delta A$. I thought $\delta$ was a scalar, but then they say that "$\delta \Lambda$ is usually not a diagonal matrix", where $\Lambda$ was diagonal. If $\delta$ is a matrix, why do we have $\Lambda + \delta \Lambda = X^{-1}(A + \delta A) X$?

$\delta$ is not a scalar, or even a variable at all here. "$\delta A$" is a single matrix-valued variable representing a perturbation to $A$.