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@pandora.be> To: "math-fun" <math-fun@mailman.xmission.com>; "Seqfan (E-mail)" <seqfan@ext.jussieu.fr> Cc: "Marc LeBrun" <mlb@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@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