Timeline for Under row operations and column permutations a matrix A can be put in the non-unique form ( I | X ), what is known about the set of possible X?

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9 events

when toggle format	what		by	license	comment
Jun 3 at 9:39	vote	accept	DeafIdiotGod
May 30 at 13:51	answer	added	Padraig Ó Catháin		timeline score: 1
May 28 at 11:29	comment	added	DeafIdiotGod		Honestly I'm having trouble finding any! The rank of $X$ is unpreserved in general, consider the case over $\mathbb{F}_2$ where $X$ is all ones except for the diagonal which is 0, this has rank 2. With $(I\|X)$ add row 1 to row 3 then swap columns 1 and 6 to obtain $(I\|X')$ where $X'$ is now rank 3.
May 24 at 16:36	comment	added	Padraig Ó Catháin		Ah - that's true. I'm not aware of any literature about this equivalence operation on rectangular matrices. Are you aware of any invariants of $X$ that are preserved?
May 24 at 16:17	comment	added	DeafIdiotGod		This is true, thank you, although I'm more interested in the $X$s that are inequivalent up to these moves. You can take a row which has a 1 in some column in the right half, add it to all other rows with a 1 in that column -- turning that column into a one-entry vector -- then normalise it and swap it to the left half to restore the $I$ block.
May 24 at 15:41	comment	added	Padraig Ó Catháin		The reduced row echelon form is unique, so you may as well start there. You can apply an arbitrary permutation of rows, apply the inverse permutation on the first block of columns to recover the identity matrix, and then apply an arbitrary permutation to the remaining columns. So I guess that $X$ is determined up to arbitrary permutation of rows and columns. Deciding whether given $X_{1}$ and $X_{2}$ are equivalent under this operation is more-or-less isomorphism of bipartite graphs.
May 24 at 13:56	history	edited	DeafIdiotGod	CC BY-SA 4.0	added 46 characters in body
S May 24 at 13:53	review	First questions
May 24 at 14:05
S May 24 at 13:53	history	asked	DeafIdiotGod	CC BY-SA 4.0