Timeline for Existence of orthogonal basis of symmetric $n\times n$ matrices, where each matrix is unitary?
Current License: CC BY-SA 4.0
21 events
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| S Dec 22, 2018 at 2:02 | history | bounty ended | CommunityBot | ||
| S Dec 22, 2018 at 2:02 | history | notice removed | CommunityBot | ||
| S Dec 14, 2018 at 0:18 | history | bounty started | Mark Girard | ||
| S Dec 14, 2018 at 0:18 | history | notice added | Mark Girard | Draw attention | |
| Dec 6, 2018 at 15:54 | comment | added | Mark Girard | @JosiahPark Unfortunately I don't think this method will work for odd $n>3$. My construction for $n=3$ works because you can "pseudo-factor" the complete graph $K_3$, as in this picture, which you can't do for odd $n\geq5$. | |
| Dec 6, 2018 at 13:14 | comment | added | Mark Girard | @JosiahPark I fixed the typo in the last line and replaced $\binom{n}{2}$ with $\binom{n+1}{2}$. Thanks. | |
| Dec 6, 2018 at 13:11 | history | edited | Mark Girard | CC BY-SA 4.0 |
fixed typo (n->n+1)
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| Dec 6, 2018 at 6:24 | comment | added | Josiah Park | For the $n=5$ case one can proceed by looking at a matrix of the form $\begin{pmatrix} \sqrt{\frac{8}{5}} & & & & \\ & \sqrt{\frac{3}{5}} & 1 & & \\ & 1 & -\sqrt{\frac{3}{5}} & & \\ & & & \sqrt{\frac{3}{5}} & 1 \\ & & & 1 & -\sqrt{\frac{3}{5}} \end{pmatrix}$ along with the matrix with flipped signs on the ones and then use the cases from $n=3$ to build up multiple unitary (up to a constant) matrices orthogonal to this matrix. Not all of the matrices in one's collection can come in pairs like this though, since the dimension of all symmetric matrices for the $n=5$ case is odd. | |
| Dec 6, 2018 at 4:58 | comment | added | Josiah Park | @luftbahnfahrer Should "if there exists a collection of n choose 2 orthogonal symmetric" be replaced with "... n+1 choose 2..." matrices near the end of the post? | |
| Dec 6, 2018 at 4:48 | comment | added | Josiah Park | @NikWeaver Sure, but these are not exact, correct? I read that they were numeric. The dimension count for $n=7$ is even like $n=3$ so it makes sense to look there possibly before $n=5$. | |
| Dec 6, 2018 at 4:46 | comment | added | Nik Weaver | @JosiahPark: according to the last part the OP has examples up to $n=11$. | |
| Dec 6, 2018 at 4:39 | comment | added | Josiah Park | This will sound dumb, but did you by chance try to construct an example with $n=7$ after $n=3$? | |
| Dec 6, 2018 at 4:07 | history | edited | Mark Girard | CC BY-SA 4.0 |
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| Dec 6, 2018 at 4:04 | comment | added | Mark Girard | Thanks, you're right! I've un-normalized all the matrices such that they are in fact unitary. | |
| Dec 6, 2018 at 4:01 | history | edited | Mark Girard | CC BY-SA 4.0 |
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| Dec 6, 2018 at 3:47 | history | edited | Mark Girard | CC BY-SA 4.0 |
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| Dec 6, 2018 at 3:36 | history | edited | Mark Girard | CC BY-SA 4.0 |
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| Dec 6, 2018 at 3:29 | history | edited | Mark Girard | CC BY-SA 4.0 |
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| Dec 6, 2018 at 0:03 | comment | added | Christian Remling | Just a trivial comment: you normalize your matrices to give them (Hilbert Schmidt) norm $1$, but of course they won't be unitary after that (none of your $U$'s is). | |
| Dec 5, 2018 at 18:47 | history | edited | Mark Girard | CC BY-SA 4.0 |
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| Dec 5, 2018 at 18:34 | history | asked | Mark Girard | CC BY-SA 4.0 |
