Let $t_{ik}$ be complex numbers. Is it true that $\max_k|\sum_i t_{ik}|=\sqrt{\max_{k,l}|\sum_i t_{ik}\overline{t_{il}}|}$?

Let $\{t_{ik}:\, i\in\{1,\ldots,p\}, k\in\{1,\ldots,m\}\}\subset\Bbb{C}$. Is it true that $\max_k|\sum_i t_{ik}|=\sqrt{\max_{k,l}|\sum_i t_{ik}\overline{t_{il}}|}$?

Let $T$ be the $m\times p$ matrix where the columns of $T$ are defined as $T_j=(t_{j1}, t_{j2},\ldots,t_{jm})^t$ and $S=TT^*$. Then, the quantity in the RHS becomes $\sqrt{\max_{k,l}S_{k,l}}$. With the help of Cauchy Schwarz inequality, this can be proved $\max_k\sum_i |t_{ik}|=\sqrt{\max_{k,l}|\sum_i t_{ik}\overline{t_{il}}|}$. But, I am unable to prove the equality when the LHS is equal to $\max_k|\sum_i t_{ik}|$.

Can anyone help me with this? Thanks for your help in advance.