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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.

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    You could start by considering the special case $B=A$. The summation is maximized at $k=\pm \infty$ (the sign depends on whether the largest eigenvalue of A is greater than 1). Does you 'objective' function has some real motivation, or is just ad-hoc?2011-11-23

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