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Let $C = A + B$, where $A$, $B$, and $C$ are positive definite matrices. In addition, $C$ is fixed. Let $\lambda (A)$, $\lambda (B)$, and $\lambda (C)$ be smallest eigenvalues of $A$, $B$, and $C$, respectively. Is there any result about the smallest eigenvalues of $C$ in comparison with the sum of smallest eigenvalues of $A$ and $B$? Is it true that : $\lambda (A)$ + $\lambda (B)$ < $\lambda (C)$ ? Moreover, what is the smallest possible value of $\lambda (A)$ + $\lambda (B)$ given a fixed $C$, and under what condition does this happen? Many thanks!

Xuan

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Question about $\lambda_{min} (A+B) > \lambda_{min} (A) + \lambda_{min} (B) $ can be seen from Weyl's inequality.

The remaining question is about the smallest attainable value of $\lambda_{min} (A) + \lambda_{min} (B) $ given a fixed $C$?

  • 0
    Are $A,B$ and $C$ symmetric? In this case you can use Rayleigh quotient $\frac{x^tCx}{x^tx}$.2012-10-09
  • 0
    A, B, and C are positive definite, therefore, symmetric. Could you clarify on how using Rayleigh quotient to find smallest possible $\lambda (A) + \lambda (B)$ ?2012-10-09

2 Answers 2