If two $n×m$ binary matrices have the same row weights, column weights, upper-diagonal weights and lower-diagonal weights, are they necessarily identical?
The weight of a given row, column, or diagonal is the number of entries with $1$ in that row, column, or diagonal.
The following matrix:$$ \begin{pmatrix} 1 & 0 & 1\\ 0 & 1 & 1\\ 1 & 1 & 0 \end{pmatrix} $$ has the row, column, and diagonal weights:
Row: $2, 2, 2$
Col: $2, 2, 2$
Upper diagonal: $1, 0, 3, 2, 0$
Lower diagonal: $1, 1, 2, 1, 1$
At first I considered only when the row and column weights are the same. I found a counter example. Then I considered only when the upper and lower diagonals weights. Yet again, I found a counter example.
How can one going about proving it? Or does anyone have a counter example? I have no been able to find one thus far.
Take$$ A = \begin{pmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 \end{pmatrix}. $$