Let $M$ be a real symmetric matrix and let $D$ be a real diagonal matrix.
Can we say something about the eigenvalues of $M+D$ in terms of the
eigenvalues of $M$?
For example, if $D$ is a constant multiple of $I$ ($D = cI$), then the eigenvalues of $M+D$ is just $c$ plus the eigenvalues of $M$. Can we get bounds on the locations of the eigenvalues of $M+D$ ?
Best Answer
I don't think anything strong can be said in general; there is no nice relationship between the eigenvalues of $M$ and $M+D$ even in the case when $M$ is $2\times 2$ or diagonal.
Obviously $M+D$ is a spectral shift (by 1) of the generalized eigenvalues of $M$ with respect to $D$, and you can prove some bounds on the minimum and maximum eigenvalues of $M+D$ in terms of those of $M$ and $D$, etc.