Periodic (-1,0,1) matrices of two types similar question: https://mathoverflow.net/questions/9547/how-to-construct-matrices-with-periodicity
Definition: a (-1,0,1) matrix is a matrix with entries either -1, 0 or 1.
I am trying to understand why (-1,0,1) (square) matrices can have any period, while (-1,0,1) matrices that are a sum of an anti-symmetric and a diagonal matrix, if periodic, seem to have periods limited to 3, 4, 6 or 12. They also seem to obey the rule that their powers are also (-1,0,1) matrices.
Brute force counting gave OEIS A072148, A111749 and A111485, but I still fail to understand the limitation on period and the confinement to (-1,0,1) of the matrix powers.
Random sampling of 7 by 7 matrices did not provide a single counter-example.
I would appreciate a link to relevant (not too high level) literature, or a nice hint. Maybe even a counter-example?
 A: For the record, and with apologies for self-answering:
the number of periodic -1,0,1 square matrices that are a sum of an anti-symmetric and a diagonal matrix is extemely limited if we regard
(1) matrices that can be obtained by a row and column permutation as equivalent (say switching row u and v and column u and v) and
(2) matrices that can be reduced to block-diagonal form by (1) as redundant (reducible).  
What remains are just 2 (trivial) 1x1-matrices (1 with period p=1, 1 with p=2), 3 2x2-matrices (2 with p=3, 1 with p=4), 5 3x3-matrices (p=4) and 2 4x4-matrices (p=4). None of the 5x5 or 6x6-matrices turn out to be irreducible, cfr (2). The sampling of the 7x7-matrices did not provide any irreducibles either. If any other irreducibles exist, they must have n >= 7.
Their periods being 1, 2, 3 or 4 evidently leads to block-diagonal composites with periods being the lcm of sets composed of [1,2,3,4] giving periods 3,4,6,12 (apart from the nxn identity matrix and its negative having p=1 or 2).  
Encoded as lower triangular nxn-matrices in concatenated form with n(n+1)/2 entries, I got
n=1;p=1  :  +
n=1;p=2  :  -
n=2;p=3  :  --0 ; -+0
n=2;p=4  :  0-0
n=3;p=4  :  --+0-- ; --+0+- ; --++0+ ; -0-+++ ; -+++0+
n=4;p=4  :  --+-0+0-+- ; --+0--+0++  
where the first entry on the penultimate line stands for the matrix
-1,+1, 0
-1,+1,+1
 0,-1,-1 
PS : by lack of alternative, operation (1) was implemented as a brute-force generation of all n! permutations and sorting their ternary representations as integers. Operation (2) was done in a similar way.
