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I search for a matrix with matrix elements $a_{im}$ that match the following rule: $$ \sum_m a_{im}\,a_{jm}\,a_{km}\,a_{lm} = \begin{cases} \alpha \quad\text{if } i=j=k=l \\ \beta \quad \text{if }i=j\neq k=l \text{ or } i=k\neq j=l \text{ or } i=l\neq j=k \\ 0 \quad \text{otherwise} \end{cases} $$ Here $\alpha>0$ and $\beta>0$ are positive real numbers. The case $\alpha=\beta$ is allowed.

I know that if I use random numbers of the intervall $[-1,1]$ as matrix elements, the relation is approximately fulfilled if the matrix has many columns. This follows from the fact that the expectation value for a matrix element $$ is zero whereas for $$ and $$ it is larger than zero.

However, isn't it possible to create a matrix that exactly matches this relation? Or at least a matrix that approximately fulfilles this rule (but without random numbers that require a large number of columns)

I didn't make any further restrictions for the matrix, so it can be complex or whatever and I can replace individual $a_{im}$ with $a_{mi}$ or $a_{mi}^*$ in the rule. It doesn't matter.

Since an orthogonal (or unitary) matrix fulfills a similar law, I have the feeling that it should be possible...

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    If $a$ is a $p \times q$ matrix, then it has $pq$ entries. Your rule consists of $p^4$ different equations. Just on general principles, there will likely be a continuum of solutions if the number of unknowns $pq$ is greater than the number of equations $p^4$, a finite number of isolated solutions if the two counts are equal, and no solutions if the number of equations is greater than the unknowns ("likely" because things occasionally go wrong in all three cases). So you will generally need $q \ge p^3$ for solutions to exist. – 2017-02-18
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    Thank you for this clarification. Ok, but is there some way to construct the matrix in a systematic way such that it obeys the rule approximately, even for $q – 2017-02-18

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