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I'm a physicist who's not particularly good at linear algebra so please accept my apologies if this is standard textbook stuff that I'm just unaware of.

I have two real rectangular matrices $A_{mxn} B_{mxn},$ where $m>n$ whose entries are $\pm1,0$. As a concrete example, take

$$ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & -1 & 1 \\ 1 & 0 & 1 \end{bmatrix}, B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & -1& 0 \\ 1 & 0 & 1 \end{bmatrix} $$ My goal is to find matrices D, P, M such that B = DPAM, where P is a permutation matrix and D is a diagonal matrix with entries $\pm1$. For example, with A, B as above I can write $$ B = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} A \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Essentially I want to find a linear transformation M such that when I form the product X=BM I obtain a matrix X which is just a row-permutation P of A. I also want to allow for the possibility that rows in PX will differ by an overall minus sign from the rows in A, so I introduce this diagonal matrix D.

In fact, I am not interesting in knowing the explicit details of the transformation (i.e. M). Given two matrices A,B I want to know whether the transformation exists and if it does, what is the determinant of D.

I'm a bit stumped so any help would be greatly appreciated.

djangology
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  • Two questions: are there restrictions on M? And from $B=DPAM$ and your requirement for X as $X=BM$ it follows $X=DPAM^2$ and you want at the same time that $X=PA$ - then this requires $ PA = DPAM^2$ where then $ A^{-1}P^{-1}D^{-1}PA = M^2$ which is an eigendecomposition of $M^2$. Did you really mean this? Or did I misread something? – Gottfried Helms Feb 27 '14 at 20:06
  • just a note that the second matrix in the second display is not a permutation matrix. there are two 1's in the last column. – Maesumi Apr 14 '19 at 19:43

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