# How to check which rows and columns of a matrix changed sign

Suppose I have two matrices $$A, B \in \mathbb{R}^{n \times n}$$. Assume that there is a way to change the signs of rows and columns of $$A$$ such that the resulting matrix is equal to $$B$$. In other words, assume that there exist diagonal matrices $$D_1, D_2 \in \mathbb{R}^{n \times n}$$ such that:

$$D_1AD_2 = B$$

where the diagonal entries of $$D_1$$ and $$D_2$$ are $$\pm 1$$.

Then is there an (efficient) algorithm to determine which rows and columns of $$A$$ changed sign to equal $$B$$?

Perhaps, some information could be retrieved from the entry-wise product of the $$\pm 1$$-sign matrix of $$A$$ and $$B$$, though I am not sure where to go from there.

Here's an approach that works if $$A,B$$ have no zero entries. Let $$M$$ denote the matrix whose entries satisfy $$m_{ij} = \begin{cases} 1 & \operatorname{sgn}(A_{ij}) = \operatorname{sgn}(B_{ij})\\ -1 & \operatorname{sgn}(A_{ij}) \neq \operatorname{sgn}(B_{ij}) \end{cases}$$ where sgn denotes the sign (AKA signum) function. Note that $$D_1AD_2 = A \odot D_1(ee^T)D_2$$ where $$\odot$$ denotes the Hadamard product and $$e = (1,\dots,1)^T$$. With that, we see that we're looking for matrices $$D_1,D_2$$ such that $$D_1(ee^T)D_2 = M$$.

We are given that such matrices $$D_1,D_2$$ exist, which means that $$M$$ is necessarily of rank $$1$$. It follows that $$M$$ can be written in the form $$uv^T$$. Because all entries of $$M$$ are equal to $$\pm 1$$, we can take $$u,v$$ to be matrices whose entries are $$1$$ or $$-1$$.

From there, we see that $$u = \operatorname{diag}(u)$$ and $$v = \operatorname{diag}(v)$$, which means that we can simply take $$D_1 = \operatorname{diag}(u)$$ and $$D_2 = \operatorname{diag}(v)$$.