Timeline for How many orthogonal matrices (not orthonormal) are there with entries in $\{0,1,−1\}$?
Current License: CC BY-SA 4.0
10 events
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Jan 27, 2022 at 17:35 | comment | added | Peter Taylor | By Paley's constructions, if $p$ is an odd prime then there's a Hadamard matrix of order $2(p + 1)$. Then Merchant's thesis says that there are $2^{\Omega(p)}$ non-isomorphic Hadamard matrices of order $4(p + 1)$. If we consider that there are $\Theta(\tfrac{n}{\lg n})$ primes for which $\tfrac n2 \le 2(p+1) \le n$ then we get an easy $2^{\Omega(n \lg n)}$ lower bound without using the Hadamard conjecture. | |
Jan 27, 2022 at 6:57 | comment | added | Arun | @Gerry I do count such matrices. | |
Jan 26, 2022 at 22:37 | comment | added | Gerry Myerson | $$\pmatrix{1&1&1&1\cr1&-1&0&0\cr0&0&1&-1\cr1&1&-1&-1\cr}$$ is nonsingular, and all rows are orthogonal, but the columns are not orthogonal. Do you count this as an orthogonal matrix? | |
Jan 26, 2022 at 19:38 | history | edited | Arun | CC BY-SA 4.0 |
clarification of the comment on non-singularity is given.
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Jan 26, 2022 at 19:37 | comment | added | Arun | @Dustin. Thanks for some useful information. I am looking only for non-singular matrices. I have edited the question. | |
Jan 26, 2022 at 3:07 | comment | added | Gerry Myerson | Do you count the zero vector as being orthogonal to other vectors? | |
Jan 25, 2022 at 22:28 | comment | added | Dustin G. Mixon | @aorq has privately indicated to me that you can get $e^{\Omega(\sqrt{n})}$ without the Hadamard conjecture by taking matrices whose rows have disjoint support. (In fact, you can restrict to nonnegative entries.) This assumes you allow rows with all zeros. | |
Jan 25, 2022 at 17:43 | comment | added | Padraig Ó Catháin | There are known to be exponentially many (in $n$) equivalence classes of Hadamard matrices at orders of the form $2^{n}$. This is a result of Eric Merchant. | |
Jan 25, 2022 at 16:40 | comment | added | Dustin G. Mixon | Assuming the Hadamard conjecture, there are $e^{\Omega(\sqrt{n})}$ inequivalent block-diagonal matrices of this form with $4k\times 4k$ and $1\times 1$ blocks on the diagonal: en.wikipedia.org/wiki/Partition_(number_theory)#Asymptotics | |
Jan 25, 2022 at 14:03 | history | asked | Arun | CC BY-SA 4.0 |