# Algebra of block matrices with scalar diagonals

I am interested in block matrices $$A$$, that is $$A\in M_{n\times n}(R)$$ where $$R=M_{s\times s}(k)$$ and $$k$$ is a field, such that for every positive integer $$m$$ the matrix $$A^m$$ has only scalar blocks on the diagonal.

If we view $$A$$ as an $$ns\!\times\! ns$$ matrix – has the characteristic polynomial of $$A$$ be of the form $$p(x)^s$$, where $$p(x)$$ is a degree $$n$$ polynomial?

Essentially the only ways I could construct such a matrix is:

1. take a Quaternion Hermitian matrix or
2. start with a block diagonal matrix with scalar blocks on the diagonal and conjugate it by a generalized Hadamard matrix $$H$$ (the $$s\times s$$ blocks of $$H$$ are orthogonal matrices and $$H*H^T=nI$$). I would expect that there should be other methods.

Is there any literature on such matrices available?