I'm working through a proof on The Hadamard determinant problem which can be found in
I don't understand how the transition from real valued matrices $A$ with entries in $\{-1,1\}$ to matrices $B = A^T A$ is justified. Of course, in the end we want to use the spectral theorem for symmetric matrices. Furthermore, $|\det A| = \sqrt{\det B}$.
But this is what the authors are writing, and I am not fully understanding:
Since multiplication of a column of $A$ by $-1$ turns $\det A$ into $-\det A$, we see that the maximum problem for $\det A$ is the same as for $\det B$.
Maybe someone unterstands the role of the multilinearity of $\det$ in this argument and can share some insight. The conclusion, which is represented in bold letters, is the part which is bothering me.