Number of points on a complex sphere with pairwise inner product restriction - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:01:24Z http://mathoverflow.net/feeds/question/77655 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77655/number-of-points-on-a-complex-sphere-with-pairwise-inner-product-restriction Number of points on a complex sphere with pairwise inner product restriction unknown (google) 2011-10-10T03:01:55Z 2011-10-10T15:31:58Z <p>Considered the following inner products:</p> <p>$(1)$ $\langle x,y \rangle = \sum_{t=1}^{n}x_{t}y_{t}$</p> <p>$(2)$ $\langle x,y \rangle_{c} = \sum_{t=1}^{n}x_{t}\bar{y}_{t}$</p> <p>consider the following surfaces:</p> <p>$\underline{Surface (a)}$: $\langle x, x \rangle = 1$</p> <p>$\underline{Surface (b)}$: $\langle x, x \rangle = \mathbb{i} = \sqrt{-1}$</p> <p>$\underline{Surface (c)}$: $\langle x, x \rangle_{c} = 1$</p> <p>In each of the above surfaces, how many points can one place so that the inner product (defined in both $(1)$ and $(2)$) between any pair of the points is purely imaginary of form $0 + \mathbb{i}r$ where $\mathbb{i}=\sqrt{-1}$ and $r \in \mathbb{R}$ and how many points are there so that the pairwise product is purely real of form $r \in \mathbb{R}$?</p> <p>The case when we seek the pairwise inner product(both $(1)$ and $(2)$ to be purely real is infinite for surfaces $(a)$ and $(c)$ (Just restrict your sphere to have purely real coordinates and search among those points).</p> <p>Likewise, the case when we seek the pairwise inner product(both $(1)$ and $(2)$ to be purely imaginary is infinite for surface $(b)$ (Just restrict your sphere to have purely imaginary coordinates and search among those points).</p> <p>What happens in the following combinations?</p> <p>$\underline{A}$:$(b)$ when we seek pure imaginary inner products (both $(1)$ and $(2)$).</p> <p>$\underline{B}$:$(a)$ and $(c)$ when we seek pure real inner products (both $(1)$ and $(2)$).</p> <p>$\underline{A}$ has been shown to have finitely many points ($O(n)$ atmost) by unknown(google) below.</p> http://mathoverflow.net/questions/77655/number-of-points-on-a-complex-sphere-with-pairwise-inner-product-restriction/77656#77656 Answer by unknown (google) for Number of points on a complex sphere with pairwise inner product restriction unknown (google) 2011-10-10T03:22:03Z 2011-10-10T03:22:03Z <p>It's easy to see you can't have infinitely many points: there would be two that are within $\epsilon>0$ of each other, and thus would have inner product very close to $\langle z,z\rangle=1$ (which would then not be purely imaginary).</p> <p>Since $\Re\langle u,v\rangle$ forms a genuine inner product on $\mathbb C^n$, two vectors whose inner product is purely imaginary would be orthogonal. Thus you can have at most $2n$ such vectors. In the other direction, it's easy to see that the following is a collection of $2n$ vectors where every pairwise inner product is purely imaginary: $$(1,0,\ldots,0),(i,0,\ldots,0),(0,1,0,\ldots,0),(0,i,0,\ldots,0),\ldots$$</p>