Suppose we have a matrix $M$ that is of dimension $n * n$. Clearly $M$ has the shape of a square. Considering $M$ can be raised to any $k$-th power, and thus $M^k$ is also a $n * n$ square matrix, is it possible to construct a matrix, say $P$, that is the shape of a perfect $n$-polygon and raise it to a $k$-th power, and $P^k$ is still a polygon?
I attached an example using a square matrix (the product and dot method). I used a pentgon as an example for what multiplying a polygon matrix with equal sides should look like. I am not sure if multiplication is even possible and if so how? The examples below are examples of squaring matrices. If that can be done, so can raising to generic powers. 
