Matrices satisfying certain pair-wise constraints - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:28:50Z http://mathoverflow.net/feeds/question/77838 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77838/matrices-satisfying-certain-pair-wise-constraints Matrices satisfying certain pair-wise constraints unknown (google) 2011-10-11T16:44:48Z 2011-10-11T20:56:55Z <p>Consider given pairs of variables: $a_{ir1},a_{ir2}\in \mathbb{R}^{m \times m}$ and $a_{jr1},a_{jr2}\in \mathbb{R}^{m \times m}$, where $r \in \{1,2,\cdots,t\}$, consider the constraints:</p> <p>$\sum_{r=1}^{t}[a_{ir1}a_{jr2}+a_{ir2}a_{jr1}]=\sum_{r=1}^{t}[a_{jr1}a_{ir2}+a_{jr2}a_{ir1}]=0$ </p> <p>$\sum_{r=1}^{t}[a_{ir1}a_{ir2}+a_{ir2}a_{ir1}]=\sum_{r=1}^{t}[a_{jr1}a_{jr2}+a_{jr2}a_{jr1}]=I$ </p> <p>with $i \ne j$ and $i,j \in \{1,2,\cdots,n\}$.</p> <p>Let the smallest size of matrices such constraints as a function of $n$ and $t$ be $f(n,t)$. My primary question is how fast does $f(n,t)$ grow with $n$ and $t$? For a fixed $t$, let the growth be $f(n)[t]$. How fast does $f(n)[t]$ grow with $n$? Does $f(n,t) = O(\log^{c}{n})$ when $t=O(n^{q})$ for some $c \in \mathbb{N}$ and $\frac{1}{3} > q \in \mathbb{Q}$?</p> <p>Secondly, how do you find such matrix solutions explicitly?</p> <p>[Note: Each $a_{ijk}$ is a square matrix.]</p> <p>What I know: For $t=1$, I am fairly certain $f(n,1) = n$. For any fixed $t$, I don't think we can do better (although not sure). What happens when $t$ is allowed to grow although with $n$ although at a sub-cubic rate w.r.t $n$ is something I am interested?</p>