Effect of Arithmetic Operations on Eigenvalues of a Matrix

Question

Say I have a matrix $X$ and another Diagonal Matrix $D$. Suppose, I create a new matrix $\bar{X} = D-X$ and am able to estimate eigenvalues of this new matrix. What information would it give me about the eigenvalues of the original matrix $X$?

Answer 1

I would say none.

For instance, let $D=diag(d_1,\dots,d_n)$ and $X=diag(x_1,\dots,x_n)$ (diagonal matrices) with $a_i$ and $x_i$ different from zero. Then the eigenvalues of $C=D-X$ are $c_i=d_i-x_i$. Now, if you have only informations about the $c_i$s, it is clear that, for a fixed $i$, every decomposition of $c_i$ into the difference of two numbers different from zero is a possibility for the eigenvalues of the two original matrices. As a consequence, you cannot say much about them. For instance, suppose that $c_1=1$, then $d_1=2^n+1$ and $x_1=-2^n$ are possible eigenvalues for every $n>0$. Therefore, you need some additional information on $X$ and/or $D$.