Diagonal Perturbations of Linear Equations

[JohnSimpson]

Consider the system of linear equations (A+D)x=b, where D is a positive semidefinite diagonal matrix. Assume for simplicity that (A+D) is full rank for any D that we care about. In my particular case of interest, D has the form D = blkdiag(0,M) for some positive diagonal matrix M. So, a subset of the diagonal entries of A are being perturbed.

Are there any off-the-shelf theorems that characterize how the components of the solution x change as the elements of D change? Intuitively as M increases in size, the bottom elements of x should shrink. Coupling through off-diagonal elements of A should then also shrink the top elements a little bit.

Thanks
-John

 

[fresh_42]

No, there isn't such a thing, simply because we have no information about $A$, which makes the form of $A+D$ arbitrary. You could only perform a blockwise addition, possibly an inversion, and establish a system of differential equations, if the entries of $D$ variate. But any new size of $M$ gives you a new problem.