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I am just curious that if I want to factorize a boolean matrix, do the factorized two components have to binary as well? I know that current work does in this way. I am just wondering whether it is possible to factorize a boolean matrix into two real matrices.

Nathan
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    Do you mean factorising a matrix containing only 1's and 0's? Otherwise you would need to give a Boolean meaning to real numbers. Any non-singular real square matrix times its inverse gives a matrix containing only 1's and 0's. – Peter Aug 05 '20 at 01:00
  • Yes, I mean I want to find A and B whose product A*B is as similar to a goal matrix C as possible. I know that if C is boolean, then usually we require that A and B are both boolean. This is known as the classical boolean matrix factorization problem and widely analyzed in machine learning. I am just curious is whether A and B must be boolean if C is boolean. Can we use real number matrix A and B? – Nathan Aug 05 '20 at 04:53
  • Suppose $C$ is any square matrix. Suppose $A$ is any non-singular matrix of the same order, so $A$ has inverse $A^{-1}$. If you let $B=A^{-1}B$, then $AB=C$. This result is trivial, and there are many different possible matrices for $A$, but if $C$ is composed only of $0$'s and $1$'s it may help. It is not "nice" - more like "factorising" $5$ into $2 \times 2\frac{1}{2}$. – Peter Aug 05 '20 at 10:39

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