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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:

$\sum_{r=1}^{t}[a_{ir1}a_{jr2}+a_{ir2}a_{jr1}]=\sum_{r=1}^{t}[a_{jr1}a_{ir2}+a_{jr2}a_{ir1}]=0$

$\sum_{r=1}^{t}[a_{ir1}a_{ir2}+a_{ir2}a_{ir1}]=\sum_{r=1}^{t}[a_{jr1}a_{jr2}+a_{jr2}a_{jr1}]=I$

with $i \ne j$ and $i,j \in \{1,2,\cdots,n\}$.

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 $f(n)$. f(n)[t]$. How fast does$f(n)$f(n)[t]$ grow with $n$? Does $f(n) f(n,t) = O(\log^{c}{n})$ when $t=O(n^{q})$ for some $c \in \mathbb{N}$?mathbb{N}$and$\frac{1}{3} > q \in \mathbb{Q}$? Secondly, how do you find such matrix solutions explicitly? [Note: Each$a_{ijk}$is a square matrix.] What I know: For$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$is something I am interested? 7 added 203 characters in body Consider given pairs of variables:$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\}$, consider the constraints:$\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\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$with$i \ne j$and$i,j \in \{1,2,\cdots,n\}$. Let the smallest size of matrices such constraints as a function of$n$and$t$be$f(n)$. f(n,t)$. My primary question is how fast does $f(n,t)$ grow with $n$ and $t$? For a fixed $t$, let the growth be $f(n)$. How fast does $f(n)$ grow with $n$? Does $f(n) = O(\log^{c}{n})$ for some $c \in \mathbb{N}$?

Secondly, how do you find such matrix solutions explicitly?

[Note: Each $a_{ijk}$ is a square matrix.]

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For

Consider given pairs of variables: $a_{i1},a_{i2}\in a_{ir1},a_{ir2}\in \mathbb{R}^{m \times m}$ and $a_{j1},a_{j2}\in a_{jr1},a_{jr2}\in \mathbb{R}^{m \times m}$, where $r \in \{1,2,\cdots,t\}$, consider the constraints:

$a_{i1}a_{j2}+a_{i2}a_{j1}=a_{j1}a_{i2}+a_{j2}a_{i1}=0$ and \sum_{r=1}^{t}[a_{ir1}a_{jr2}+a_{ir2}a_{jr1}]=\sum_{r=1}^{t}[a_{jr1}a_{ir2}+a_{jr2}a_{ir1}]=0a_{i1}a_{i2}+a_{i2}a_{i1}=a_{j1}a_{j2}+a_{j2}a_{j1}=I$\sum_{r=1}^{t}[a_{ir1}a_{ir2}+a_{ir2}a_{ir1}]=\sum_{r=1}^{t}[a_{jr1}a_{jr2}+a_{jr2}a_{jr1}]=I$

with $i \ne j$ and $i,j \in \{1,2,\cdots,n\}$.

Let the smallest size of matrices such constraints as a function of $n$ be $f(n)$. My primary question is how fast does $f(n)$ grow with $n$?

Secondly, how do you find such matrix solutions explicitly?

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