Can the following conclusion hold?

There exist matrices $H^i\in\mathbb{R}^{m\times n}, (m\leq n, i\in\mathcal{N})$, and non-zero real numbers $\underline{h}^i$, and $\overline{h}^i$. If the following inequality holds \begin{align} {\underline{h}^i}^2 I_{m}\leq H^{i} (H^{i})^T \leq {\overline{h}^i}^2 I_{m} \end{align} and \begin{align} \sum_{i\in\mathcal{N}}(H^{i})^T H^{i} >0, \end{align} then \begin{align} \sum_{i\in\mathcal{N}}(H^{i})^T H^{i} \geq \min_{i\in\mathcal{N}}\{{\underline{h}^i}^2\} I_n \end{align}

Note: The expression matrix $M>0$ means matrix $M$ is positive definite. Similarly, the expression $A\geq B$ means $A-B$ is positive semi-definite.

  • $\begingroup$ What does it mean for one matrix to be less than or equal to another matrix? Is $h$ a real number? Also, in the last line, what is the minimum being taken over? $\endgroup$ – Chandler Watson Jul 21 '18 at 6:16
  • $\begingroup$ Thanks for your questions. The descriptions have been updated such that all your concerns have been addressed. $\endgroup$ – wayne Jul 21 '18 at 12:00

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