Eigenvalues of the sum of a positive and a positive semidefinite matrix

Question

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times n}$ respectively a positive and a positive semi-definite matrix. Is it possible to establish an upper bound for the minimum eigenvalue of the sum?

Answer 1

Yes. Let $\lambda_1(A)\geq \cdots \geq \lambda_n(A)$ denote the eigenvalues of $A$. Weyl's inequality gives us the bound $$ \lambda_n(A + B) \leq \lambda_i(A) + \lambda_{n+1-i}(B) \quad i = 1,\dots,n. $$ In particular, we can see that if $B$ has rank $r$, then taking $i = n-r$ gives us $$ \lambda_n(A + B) \leq \lambda_{n-r}(A). $$