About positive definite matrix If $A$ is positive definite(uniformly to $w$) $N \times N$ matrix whose components ${a_{ij}} = a_{ij} (w) $, then how can I prove that there exist constants $c_1 = c_1 (M), c_2 = c_2 (M) >0$ such that for any $\xi, w \in \Bbb C^N$,  if $|w| \leqslant M $ then  $$ c_1 | \xi|^2 \leqslant \sum_{i,j=1}^N a_{ij} (w) \bar\xi_i \xi_j \leqslant c_2 | \xi|^2 ?$$ 

The closed unit ball of $\Bbb C^N$ is compact, and so is $X:=\{(\omega,\xi),|\omega|\leq M,|\xi|=1\}$. Let $f(\omega,\xi)=\sum_{k,j=1}^Na_{kj}(\omega)\bar{\xi_k}\xi_j$. Assuming the $a_{i,j}$ continuous, and applying the result of this thread  to $X$ and this function, we get the wanted result. 
