Given positive semi-definite $n\times n$ matrices $B_k$, how would I go about getting the limits on $\alpha_k$ such that the expression \begin{equation} \mathbb{I}-\sum_{k=1}^{m-1}\alpha_kB_k \end{equation} is a positive definite matrix? In the expression above, there are $m-1$ matrices (the $B_k$ 's), all positive semi-definite. Also, there is a condition on $\alpha_k$ that \begin{equation} \sum_{k=1}^{m-1}\alpha_k<1,\quad \alpha_k>0 \end{equation}
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$\begingroup$ You can drop your condition, which just says $\sum_{k=1}^{m-1}\alpha_k\le1$ (supposing you want $\alpha_k\ge0$). So you may as well write $\mathbb{I}-\sum_{k=1}^m\alpha_kB_k $, with condition $\sum_{k=1}^{m}\alpha_k\le1$. $\endgroup$– WolfgangFeb 5, 2016 at 11:06
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