I'm trying to better understand matrices that are defined by preserving some other tensor. I recently asked a more involved question, and I realized it might be better to learn simpler examples first.
As an example, consider the set of $n$ by $n$ real matrices $M$ that satisfy $M_{ik} M_{jl} \delta_{kl} = \delta_{ik}$. Here $\delta_{ij}$ is the $n \times n$ Kronecker delta. This is merely a restatement of $M M^T = I$ which is the defining relation of orthogonal matrices. By exploring $M$ close to the identity, we can deduce that up to a reflection, $M$ will be the exponential of a real antisymmetric matrix.
I want to understand generalizations where the $M$'s instead preserve some trilinear form, not a bilinear one. Consider for example $\delta_{ijk}$, the $n\times n \times n$ tensor that is $1$ if $i=j=k$ and $0$ otherwise.
How can we characterize the set of real matrices $M$ that satisfy $M_{il} M_{jm} M_{kn} \delta_{lmn} = \delta_{ijk}$? In particular, is a complete enumeration of these $M$'s known?
By inspection, one can see that if $M_1$ and $M_2$ satisfy this constraint, then so does $M_1 M_2$. Similarly, if $M$ satisfies this constraint and is invertible, then so too does $M^{-1}$. However, I'm finding it hard to say general statements about the properties of matrices $M$ satisfying $M_{il} M_{jm} M_{kn} \delta_{lmn} = \delta_{ijk}$. For example, it's not even obvious to me whether all $M$ that satisfy this constraint must be invertible.
By inspection, we can deduce some explicit solutions. If $M$ is a permutation matrix, it will satisfy the above equation. However, I'm having trouble guessing any more real matrices that satisfy the constraint. As an aside, if I weaken the constraint and allow $M$ to be complex, then note that I get additional solutions from the set of diagonal $M$ whose cube is the identity, but again I suspect I might be missing solutions.
I suspect and hope that there are techniques that will give a complete enumeration of the kinds of matrices $M$ satisfying this constraint.