Let $A\in\mathbb{R}^{d\times d}$ and $B\in\mathbb{R}^{d\times d}$ be two positive definite matrices. $k$ is a real coefficient. Suppose the largest eigenvalue of $A-kB$ is $\lambda_1$. Is it possible to find a $k$ such that $k+\lambda_1$ is maximized?
Here is my opinion:
When $k$ is positive and large, $A-kB$ may be negtive definite such that $\lambda_1$ is negtive. When $k$ is negtive, $A-kB$ is positive definite such that $\lambda_1$ is positive. So roughly speaking, large $k$ will give small $\lambda_1$, while small $k$ will give large $\lambda_1$. When can their summation be maximized? Thanks.