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I am interested in the QL decomposition of a binary invertible matrix in $\mathbb{F}_2^{n\times n}$.

I am aware that finding the QL decomposition is equivalent to finding the QR decomposition from some old Reddit post. Consider the binary matrix $F$ with 1's in the anti-diagonal and everywhere 0. Then this is a particular instance of a permutation matrix; $AF$ reverses the order of the columns of $A$, and $FA$ reverses the order of the rows of $A$. We also have $F^2$ = 1. Then for any binary matrix $A$, once we know how to do the 'binary' QR decomposition, then we can decompose as $AF = QR$. Then, $$A = AF^2 = (AF)F = QRF = QF^2RF = (QF)(FRF)$$ in which QF is again an orthogonal matrix, and FRF is a lower triangular matrix. Hence if we let $Q'=QF$ and $L' = FRF$, then $A = Q'L'$, leading to a QL decomposition of $A$. A symmetric argument leads that the QR decomposition is equivalent to finding the QL decomposition.

This argument seems to work for binary matrices, so I need to find either QL or QR decomposition of binary matrices. Surprisingly, googling these keywords didn't seem to bring any fruitful result. How would one tackle this problem?

Edit) As I mentioned in the comments, I want to know whether the QR decomposition exists for invertible binary matrices, but I do not care whether the computation is impractical or if there is no algorithm.

Joshua Woo
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  • Isn't QR decomposition basically the same thing as Gram-Schmidt on the columns? And Gram-Schmidt does not exist in the binary world because there are plenty of zero norm vectors. – Jyrki Lahtonen Jan 19 '24 at 04:05
  • Problems like these. There are many serious obstacles to using tools based on inner products in positive characteristic. May be the root cause is that there is no subset of positive elements closed under products and addition. – Jyrki Lahtonen Jan 19 '24 at 04:12
  • @JyrkiLahtonen Hmm.. does this mean that the QR/QL decomposition is infeasible in binary space? – Joshua Woo Jan 19 '24 at 04:14
  • Symmetric bilinear forms, OTOH, are a well studied object in the binary world also. A key feature is that they cannot be diagonalized. Don't know if that is at all relevant here though? Depends on what is the problem you really want to solve. – Jyrki Lahtonen Jan 19 '24 at 04:19
  • May be QR/QL is totally infeasible in binary? I guess it depends on exactly what you are looking for. – Jyrki Lahtonen Jan 19 '24 at 04:20
  • @JyrkiLahtonen Thank you for your kind answer. I searched the Wikipedia page, and it seems that the QR decomposition can be calculated using several ways, the Gram-Schmidt process, Householder reflections and possibly some more. I want to know whether the QR decomposition exists for invertible binary matrices, but I do not care whether the computation is impractical or if there is no algorithm. – Joshua Woo Jan 19 '24 at 04:31
  • This seems a lot advanced for me. I'll try to understand it. Again, thank you for your help! – Joshua Woo Jan 19 '24 at 04:39
  • Something like the Bruhat decomposition? I'm ashamed to admit I don't remember how it plays out with finite fields (WP mentions algebraically closed fields). – Jyrki Lahtonen Jan 19 '24 at 04:41
  • Note the factor $2$ in the construction of Householder reflections – in $\mathbb F_2$, that's $0$. – joriki Jan 19 '24 at 15:30

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There are only two orthogonal binary $2\times2$ matrices, $\pmatrix{1&0\\0&1}$ and $\pmatrix{0&1\\1&0}$. Right-multiplying one of them by an upper triangular matrix yields at least one $0$ in the first column, so a binary $2\times2$ matrix with two $1$s in the first column has no QR decomposition. The matrix $\pmatrix{1&1\\1&0}$ is invertible and has two $1$s in the first column, so invertible binary matrices need not have a QR decomposition.

joriki
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