Invariant subspaces of circulant and recursive matrices

Invariant subspaces of circulant and recursive matrices are studied. It is shown that there are no invariant subspaces for circulant matrices under certain conditions, except for a specific class that has been found previously. The proposed conditions hold for maximum distance separable matrices, and as a consequence, they also hold for matrices used in the AES block cipher and the Whirlpool hash function. We propose a class of subspaces that are invariant under a parallel, similar S-box transformation. We demonstrate that there are no such invariant subspaces for recursive matrices under some conditions. In particular, the linear transformation used in the Kuznyechik block cipher has no invaruant subspaces.