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 $C\_i$.



Perturbation theory for matrices
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.


