I asked the following question on math.stackexchange.com but have not received any response. So I would like to try my luck here.
This question is related to the Grothendieck inequality. Let field $\mathbb F$ be $\mathbb R$ or $\mathbb C$ and $m,n\in \mathbb N$. What $M\in \mathbb F^{m\times n}$ would satisfy the following equality? $$ \max_{\|x_i\|=\|y_j\|=1} \sum_{i,j} M_{i,j} \langle x_i, y_j \rangle = \max_{|s_i|=|t_j|=1} \sum_{i,j} M_{i,j} s_i t_j, $$ where $\langle \cdot, \cdot \rangle$ is the inner product in a Hilbert space and $\|\cdot\|$ is the 2-norm in that space, and $|\cdot|$ is the absolute value in $\mathbb F$.
$uM$ where $|u|=1$ satisfies this equality, for any $M$ satisfying this equality. A positive matrix $M$ where $M_{i,j}\ge0,\,\forall i,j$ obviously satisfies this inequality. So does a rank $1$ matrix. What is the characterization of all such $M$?
