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I'm looking for a matrix version of the basic inequality for the ratio of two sums of positive numbers: $$\frac{a + b}{c + d} \leq \max\left\{\frac{a}{c}, \frac{b}{d}\right\}.$$ Specifically, I have positive semidefinite matrices $A, B \in \mathbb{R}^{d \times d}$ such that $(A + B)$ is invertible. I'm looking for a bound on $$\|(A + B)^{-1}(A x + B y)\|.$$ In the more general problem (the problem I actually care about), $A_i\ (i = 1, \dotsc, n)$ is a finite sequence of positive semidefinite matrices and I want a bound on $$\rho_n \equiv \Big\|\Big(\sum_{i=1}^{n} A_i\Big)^{-1}\Big(\sum_{i=1}^{n} A_i x_i\Big)\Big\|.$$

If it helps, assume that $A_i$ is a projection matrix and that $\|\cdot\|$ is Euclidean norm; I suspect that in in this case, $\rho_n \leq d^{1/2} \max_{i} \|x_i\|$.

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  • $\begingroup$ What is your norm? $\endgroup$ – Siméon Nov 27 '13 at 14:39
  • $\begingroup$ I'm mostly interested in Euclidean norm, but I'd be happy with $\|\cdot\|_1$ or $\|\cdot\|_\infty$. $\endgroup$ – Patrick Perry Nov 27 '13 at 15:04
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I have an answer for Euclidean norm. Set $\Omega = \sum_{i=1}^{n} A_i$ and let $B_i = \Omega^{-1} A_i$. Then, $$\rho_n = \big\|\sum_{i=1}^{n} B_i x_i\big\| \leq \sum_{i=1}^{n}\|B_i\| \|x_i\| \leq \big(\sum_{i=1}^{n} \|B_i\|\big) \max_i \|x_i\|.$$ Set $\tilde B_i = \Omega^{-1/2} A_i \Omega^{-1/2} \succeq 0$, noting that $\|B_i\| = \|\tilde B_i\|$. Furthermore, $\sum_{i=1}^{n} \tilde B_i = I$, so $$\sum_{i=1}^{n}\|\tilde B_i\| \leq \sum_{i=1}^{n} \mathrm{tr}(\tilde B_i) = d.$$ Thus, $$\rho_n \leq d \max_i \|x_i\|.$$ I don't think that this is tight, but it works fine for my purposes.

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