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

wouter meeussen wouter.meeussen at pandora.be
Sat Nov 15 20:01:39 CET 2003

```One counter-example to spoil things a bit:

for n=7, a small sample of (-1,0,1)-matrices whose powers are 'same',
shows that not all are pseudo-Antisymmetric (= AntiSymmetric + Diagonal)

try:
{
{-1,  0,  0, 1, 0,  0,  0},
{ 0, -1,  0, 0, 0,  0,  0},
{ 0,  0,  0, 0, 0,  1, -1},
{-1,  0,  0, 0, 0,  0,  0},
{ 0,  0,  0, 0, 1,  0,  0},
{ 0,  0, -1, 0, 0, -1,  1},
{ 0,  0, -1, 0, 0, -1,  0}
}

so my program might have missed some for n=4 or n=5,
since I examined only the pseudo-antisymmetric ones.

But the (powerlength = divisor of 12) remains valid.

When interpreted as adjacency-matrices of (oriented?) graphs,
do their matrix-powers have a 'graph-theoretical meaning'?

Wouter.

----- Original Message -----
From: "wouter meeussen" <wouter.meeussen at pandora.be>
To: "math-fun" <math-fun at mailman.xmission.com>; "Seqfan (E-mail)" <seqfan at ext.jussieu.fr>
Cc: "Marc LeBrun" <mlb at fxpt.com>
Sent: Saturday, November 08, 2003 7:27 PM
Subject: moRe: powers of PseudoAntisymmetric (-1,0,1)- Matrices

> 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
>
>
>
>

```