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?