# Comparing binary matrices and weights

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.

• What do these weights even mean? – darij grinberg Mar 16 '18 at 0:43
• Pigeonhole proof: there are $2^{n^2}$ such matrices, but at most $(n+1)^{6n}$ such configurations of weights ($n+n+2n+2n$ different weights, each of which takes a value in $\{0,\dots,n\}.$ The $2^{n^2}$ is bigger for sufficiently large $n.$ – Dap Mar 16 '18 at 9:01

Take$$A = \begin{pmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0 \end{pmatrix}.$$