Could someone help me to prove or disprove the following inequality?


Let $(c_{nr})$ be an $Ntimes R$ complex matrix, then $forall z_n in mathbb{C}$, we have
$$
sum_r |sum_n c_{nr}z_n|^2 geq frac{1}{sigma_{max}} sum_n |z_n|^2
$$

where $sigma_{max}$ is the maximal sigular value of the complex matrix $(c_{nr})$.