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We have $N$ Hermitian matrices $A_i$ and $N$ weight values $w_i$, $1\leq i\leq N$, $\sum_{i=1}^N w_i=1$.

Then we can obtain a new Hermite matrices $\sum_{i=1}^N w_iA_i$. let us assume $\lambda$ is the minmum non-zero eigenvalue of $\sum_{i=1}^N w_iA_i$, and vector $X$ is the corresponding eigenvector.

My question is how to select $w_i$， so that $\max_iX^HA_iX$-$\min_iX^HA_iX$ is as minimal as possible.

thanks for your answer.

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  • $\begingroup$ I guess you mean Hermitian matrices? $\endgroup$
    – Suvrit
    Aug 28, 2012 at 16:46
  • $\begingroup$ Sorry, its a mistake, $A_i$ is a Hermitian matrix. Thank you. $\endgroup$
    – leslie
    Aug 29, 2012 at 7:17

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