My question arises from a discussion on an answer given by Maurizio Monge here.I do not know if there is a known terminology for such matrices. By "sign matrices," I mean square matrices whose entries are in ${-1,+1}$.
For instance,
$\begin{bmatrix}
1 &-1 \\
-1& -1
\end{bmatrix}$ ,
$\begin{bmatrix}
-1&1&1 \\
1&1&-1 \\
-1&-1&-1
\end{bmatrix}$
Clearly, there are $2^{n^2}$ sign matrices of size $n\times n$. So, we start their theory by enumerating them as follows. For a matrix of size $n\times n$ we consider a truth table of $n^2$ arguments and therefore $2^{n^2}$ rows. Each row corresponds to the entries in one matrix$(a_{11},a_{12},\dots,a_{1n},a_{21},a_{22},\dots,a_{nn})$.
Let $M_{(n,k)}$ be the $n \times n$ sign matrix corresponding to the $k^th$ row of the truth table.
Question: Does the following matrix product give the zero matrix for sign matrices of even size?
$\prod_{k=1}^{2^{n^2}}M_{(n,k)}$
Thank you. As usual, I will be delighted if you point me to good references on this.
Best Answer
Much is known about sign nonsingular patterns (sign patterns for which nonsingularity does not depend on the numerical values), if I remember correctly there is a characterization. Less is known about sign patterns which have allow (but do not require) nonsingularity. I suggest looking at the book Matrices of sign-solvable linear systems by Brualdi and Shader.