Perturbation theory for matrices

Question

I encountered the following problem. Since this is somewhat not related to what I normally do, I wanted to know what the best estimates in this field are.

Let $A \in \mathbb{R}^{n \times n}$ be a self-adjoint matrix and $B:=A + \sum_{i=1}^{k} w_i C_i$ where $w_i \in \mathbb{R}$ and $C_i$ an orthonormal system of self-adjoint matrices with positive and negative eigenvalues. I would like to know now if there is a better estimate on the lowest eigenvalue of $B$ then this one? $\lambda_1(B) \ge \lambda_1(A) + \sum_{i=1}^{k} w_i \kappa_i$ where $\kappa_i=\lambda_1(C_i)$ if $w_i$ is positive, or $\kappa_i= \lambda_n(C_i)$ if $w_i$ is negative. Here $\lambda_1(A)$ is the smallest eigenvalue of $A$ and $\lambda_n(A)$ the largest one.

-This is the most simple one I can think of, but it does for example not take into account the orthonormality of the $C_i$.

Answer 1

By orthonormal, I suppose you mean $\text{tr}(C_i C_j) = 0$ for $i \ne j$, $\text{tr}(C_i^2) = 1$?

The estimate you gave is tight, in the sense that it is an equality if, for example, all $w_i = \epsilon > 0$ sufficiently small and the matrices $A$ and $C_i$ all share the same eigenvector for the lowest eigenvalue, this eigenvalue for $A$ being simple. There are examples of this with the $C_i$ diagonal matrices.