# moRe: powers of PseudoAntisymmetric (-1,0,1)- Matrices

wouter meeussen wouter.meeussen at pandora.be
Sat Nov 8 19:27:31 CET 2003

```rehash of thread 15/08/2003, with some news.
OEIS?Anum=A072148   Sequence:  2,14,92,796,7672

Definitions:
pseudoAntisymmetric : T(i,j)= -T(j,i) for j<i , so T = diagonal+Antisymmetric.
(my definition, forgive..)

powerlength: minimal p>0 so that T^p = Identity

Consider
the (-1,0,1)-matrices T with properties : Det[T] not zero (invertible),
all powers T^k are also invertible (-1,0,1) matrices.

Properties:
powerlength of T divides 12,
Det[t] is 1 or -1,
T is pseudoAntisymmetric,

the powers T^k need not be all pseudoAntisymmetric:

for 4x4 matrices,
all those with 8 non-zero elements have
powerlength 4, and their powers 2 and 3 are not pseudoAntisymmetric;

for the 5x5 matrices,
all those with 9 non-zero elements have
powerlength 4, and their powers 2 and 3 are not pseudoAntisymmetric;
all those with 10 non-zero elements have
powerlength 12, and their powers 2,3,6,7,10 and 11 are not pseudoAntisymmetric;

There is a system in this madness,
but this margin is too small...

W.

(I owe Marc LeBrun <mlb at fxpt.com> for help,
partial insight & lots inspiration, thanx Marc)

I put the 796 4-by4 and the 7672 5-by-5 on
http://users.pandora.be/Wouter.Meeussen/pseudoAntisymmMatrixPowers_4.txt
http://users.pandora.be/Wouter.Meeussen/pseudoAntisymmMatrixPowers_5.txt

```