most recent 30 from http://mathoverflow.net 2013-06-18T06:27:26Z http://mathoverflow.net/feeds/question/105678 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105678/a-weighted-sum-of-hermitian-matrices-and-selection-of-weight-values leslie 2012-08-28T03:28:50Z 2012-08-29T07:18:31Z

<p>We have $N$ Hermitian matrices $A_i$ and $N$ weight values $w_i$, $1\leq i\leq N$, $\sum_{i=1}^N w_i=1$.</p> <p>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.</p> <p>My question is how to select $w_i$， so that $\max_iX^HA_iX$-$\min_iX^HA_iX$ is as minimal as possible.</p> <p>thanks for your answer. </p>