MatrixPower works inconsistently for a matrix of elements of GF[p]

• To: mathgroup at smc.vnet.net
• Subject: [mg122329] MatrixPower works inconsistently for a matrix of elements of GF[p]
• From: Francois Grieu <fgrieu at gmail.com>
• Date: Tue, 25 Oct 2011 06:18:16 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```Hello,

I'm trying to evaluate powers of a square matrix of elements in GF[p].

I obtain mostly the correct result with MatrixPower, as follows.
However, some of the elements equal to 0 (mod p) are plain 0,
rather than elements of GF[p]; how could that be?

<<FiniteFields`;
m={
{GF[5][{1}], GF[5][{1}]},
{GF[5][{1}], GF[5][{0}]}
};
For[n=1,n<12,n++,Print["n=",n," ",MatrixPower[m,n]]];

gives:

n=1 {{Subscript[{1}, 5],Subscript[{1}, 5]},{Subscript[{1}, 5],Subscript[{0}, 5]}}
n=2 {{Subscript[{2}, 5],Subscript[{1}, 5]},{Subscript[{1}, 5],Subscript[{1}, 5]}}
n=3 {{Subscript[{3}, 5],Subscript[{2}, 5]},{Subscript[{2}, 5],Subscript[{1}, 5]}}
n=4 {{0,Subscript[{3}, 5]},{Subscript[{3}, 5],Subscript[{2}, 5]}}
n=5 {{Subscript[{3}, 5],0},{Subscript[{0}, 5],Subscript[{3}, 5]}}
n=6 {{Subscript[{3}, 5],Subscript[{3}, 5]},{Subscript[{3}, 5],0}}
n=7 {{Subscript[{1}, 5],Subscript[{3}, 5]},{Subscript[{3}, 5],Subscript[{3}, 5]}}
n=8 {{Subscript[{4}, 5],Subscript[{1}, 5]},{Subscript[{1}, 5],Subscript[{3}, 5]}}
n=9 {{0,Subscript[{4}, 5]},{Subscript[{4}, 5],Subscript[{1}, 5]}}
n=10 {{Subscript[{4}, 5],0},{0,Subscript[{4}, 5]}}
n=11 {{Subscript[{4}, 5],Subscript[{4}, 5]},{Subscript[{4}, 5],Subscript[{0}, 5]}}

Also, the power n in MatrixPower[m,n] is limited: MatrixPower[m, 2^31] fails,
although only 31 matrix multiplications are required to evaluate it; how comes?

TIA,

Francois Grieu

```

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