Let $M$ be an invertible symmetric $2n \times 2n$ matrix with entries in the finite field $\mathbb{F}_2$. Is $\mathrm{Ker}\ (M^2 - I_{2n})$ necessarily even dimensional?
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$\begingroup$ Where does this come from? Looks a lot like homework. $\endgroup$– Igor RivinCommented Mar 11, 2015 at 0:50
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4$\begingroup$ It's not homework. If $A$ is a principally polarized abelian variety over a finite field $\mathbb{F}_q$ of odd characteristic, is $A(\mathbb{F}_{q^2})[2]$ even dimensional? $\endgroup$– Lisa S.Commented Mar 11, 2015 at 0:53
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3$\begingroup$ $\left(\begin{array}{clcr}1&1&0&0\\1&1&1&0\\0&1&1&0\\0&0&0&1 \end{array} \right).$ $\endgroup$– Geoff RobinsonCommented Mar 11, 2015 at 1:12
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2$\begingroup$ In this context $A$ must be in ${\rm Sp}_{2n}$, yes? $\endgroup$– Noam D. ElkiesCommented Mar 11, 2015 at 2:16
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1$\begingroup$ The condition that $M$ be in ${\rm Sp}(2n,2)$ is that $M^{t}\left(\begin{array}{clcr} 0 & I_{n}\\I_{n} & 0 \end{array}\right)M = \left(\begin{array}{clcr} 0 & I_{n}\\I_{n} & 0 \end{array}\right).$ $\endgroup$– Geoff RobinsonCommented Mar 11, 2015 at 20:10
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2 Answers
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The matrix $M = \left(\begin{array}{clcr} 1&1&0&0\\1&1&1&0\\0&1&1&0\\0&0&0&1\end{array} \right)$ is an example where the dimension of the space in question is odd.
(Later edit: However if $M = M^{t}$ and also $M \in {\rm Sp}(2n,2),$ then we in fact have $M^{2} = I_{2n}$ as Darij Grinberg and Noam Elkies implicitly noted in comments).
answered Mar 11, 2015 at 7:56
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As Geoff Robinson has already said, the answer to the question is no. In dimension $4$, there are in total $120$ counterexamples, of which $96$ have kernel of dimension $1$, and $24$ have kernel of dimension $3$. This can be found out e.g. with GAP as follows:
gap> solutions := Filtered(AsList(GL(4,2)),
> M -> M = TransposedMat(M) and
> Length(NullspaceMat(M^2-One(M))) mod 2 = 1);;
gap> Length(solutions);
120
gap> Collected(List(solutions,M->Length(NullspaceMat(M^2-One(M)))));
[ [ 1, 96 ], [ 3, 24 ] ]
A list of these $120$ counterexamples is as follows (dots denote zeros):
. . . 1 1 . 1 . 1 . . . 1 . . . . 1 . . 1 . . . 1 1 . 1 . 1 1 .
. 1 1 1 . . 1 . . . 1 . . . 1 . 1 . . 1 . 1 1 1 1 . . 1 1 . . 1
. 1 . 1 1 1 . . . 1 1 1 . 1 . 1 . . 1 . . 1 . . . . . 1 1 . 1 .
1 1 1 1 . . . 1 . . 1 1 . . 1 1 . 1 . 1 . 1 . 1 1 1 1 1 . 1 . .
1 . . . 1 . . . 1 . . 1 1 . . 1 . 1 . . 1 . . 1 . 1 1 . 1 1 . .
. . 1 1 . . 1 1 . 1 . . . 1 . . 1 1 1 1 . 1 . . 1 1 . . 1 . . 1
. 1 1 1 . 1 . . . . . 1 . . . 1 . 1 . 1 . . 1 1 1 . . 1 . . 1 .
. 1 1 . . 1 . 1 1 . 1 . 1 . 1 1 . 1 1 1 1 . 1 1 . . 1 . . 1 . .
1 1 1 . . 1 . . . 1 . 1 1 . . 1 . 1 . . . . 1 . . 1 1 . 1 . . .
1 1 . . 1 . 1 . 1 . 1 . . . 1 . 1 1 1 . . . 1 1 1 1 1 . . 1 . 1
1 . 1 . . 1 1 . . 1 1 . . 1 . 1 . 1 1 . 1 1 . . 1 1 . . . . . 1
. . . 1 . . . 1 1 . . . 1 . 1 . . . . 1 . 1 . 1 . . . 1 . 1 1 .
1 . . . 1 . . . . 1 . . . 1 . 1 1 . 1 1 1 1 1 1 1 . 1 . . 1 1 .
. 1 . 1 . 1 1 . 1 1 . 1 1 . . 1 . 1 . . 1 . . . . 1 . . 1 . . .
. . . 1 . 1 1 1 . . 1 . . . 1 . 1 . . . 1 . . 1 1 . 1 1 1 . 1 .
. 1 1 1 . . 1 . . 1 . 1 1 1 . 1 1 . . 1 1 . 1 1 . . 1 . . . . 1
1 . . . 1 1 . . 1 . 1 . 1 1 1 . 1 1 1 1 1 . 1 1 . 1 1 . . 1 . .
. 1 1 . 1 . 1 . . . . 1 1 1 1 1 1 . 1 . . . 1 . 1 1 1 . 1 . 1 .
. 1 1 1 . 1 . . 1 . . 1 1 1 . . 1 1 1 . 1 1 1 1 1 1 1 1 . 1 . 1
. . 1 1 . . . 1 . 1 1 . . 1 . . 1 . . . 1 . 1 . . . 1 . . . 1 1
. 1 . 1 . . 1 1 1 . 1 . . . 1 1 . 1 . 1 1 1 1 . . . . 1 . 1 . 1
1 1 . 1 . 1 . . . 1 . . . 1 . . 1 1 . . 1 1 . . . 1 . . 1 1 . .
. . 1 . 1 . 1 . 1 . . 1 1 . . 1 . . 1 . 1 . . . . . 1 1 . . . 1
1 1 . . 1 . . . . . 1 . 1 . 1 1 1 . . . . . . 1 1 . 1 1 1 . 1 .
. . 1 1 . . 1 . . . . 1 . . 1 . 1 1 . 1 1 1 1 . . 1 . 1 1 1 . .
. . . 1 . 1 1 . . 1 . 1 . 1 1 1 1 . . 1 1 . . . 1 . 1 . 1 . . 1
1 . 1 1 1 1 1 . . . 1 . 1 1 1 1 . . 1 . 1 . 1 . . 1 . . . . . 1
1 1 1 1 . . . 1 1 1 . 1 . 1 1 . 1 1 . . . . . 1 1 . . 1 . 1 1 .
1 . 1 1 1 1 . 1 1 . 1 1 1 1 . 1 1 1 . 1 1 . 1 . 1 . 1 . 1 . . 1
. 1 . . 1 . . . . 1 . . 1 1 . . 1 1 . . . 1 . . . . 1 . . . 1 1
1 . 1 . . . 1 . 1 . 1 . . . 1 . . . 1 . 1 . 1 1 1 1 1 . . 1 . .
1 . . 1 1 . . 1 1 . . . 1 . . 1 1 . . . . . 1 1 . . . 1 1 1 . .
. 1 . 1 1 1 1 . 1 1 . . . 1 . . 1 . . . . . 1 1 1 1 1 . . 1 . 1
1 . . . 1 . 1 . 1 1 . 1 1 1 1 1 . 1 . 1 . . . 1 1 . 1 . 1 . . .
. . 1 1 1 1 1 1 . . 1 . . 1 1 1 . . 1 1 1 . 1 . 1 1 . . . . 1 .
1 . 1 . . . 1 . . 1 . 1 . 1 1 . . 1 1 1 1 1 . . . . . 1 1 . . 1
1 1 . 1 1 1 . . . . 1 . 1 1 1 1 . 1 . . 1 . . . 1 . . . 1 . . 1
1 1 1 1 1 1 . 1 . 1 . 1 1 1 . 1 1 . . 1 . . 1 1 . . 1 1 . 1 . 1
. 1 . . . . 1 . 1 . . 1 1 . . . . . 1 1 . 1 1 . . 1 . 1 . . 1 .
1 1 . . . 1 . . . 1 1 . 1 1 . . . 1 1 . . 1 . . . 1 1 1 1 1 . 1
1 . . . 1 . . . . . 1 1 . . . 1 . 1 1 . 1 1 1 1 1 . . . 1 1 . .
. . . 1 . . . 1 . 1 . 1 . . 1 1 1 . 1 . 1 . . 1 . 1 1 1 1 1 1 .
. . 1 1 . . 1 1 1 . . . . 1 1 1 1 1 1 . 1 . . . . 1 . 1 . 1 1 .
. 1 1 . . 1 1 1 1 1 . . 1 1 1 1 . . . 1 1 1 . 1 . 1 1 . . . . 1
1 . . . 1 . . . . . 1 . 1 . 1 1 . 1 1 . . . 1 . . . 1 . . . . 1
. 1 1 1 . 1 1 1 . 1 . . . 1 . . 1 1 . . . 1 . . . 1 1 . . . 1 1
. 1 1 . . 1 1 . 1 . . 1 1 . . 1 1 . . . 1 . 1 1 1 1 . . . 1 1 .
. 1 . . . 1 . 1 . . 1 1 1 . 1 . . . . 1 . . 1 1 . . . 1 1 1 . .
. . 1 1 . . 1 1 . 1 1 . 1 1 . . . . . 1 1 1 1 1 . . . 1 . 1 . 1
. 1 1 . . 1 . . 1 . . . 1 1 1 . . 1 . 1 1 1 1 . . 1 1 . 1 1 1 1
1 1 . . 1 . 1 1 1 . . 1 . 1 . . . . 1 . 1 1 . . . 1 . 1 . 1 . .
1 . . . 1 . 1 . . . 1 1 . . . 1 1 1 . . 1 . . . 1 . 1 . 1 1 . 1
. . 1 1 . 1 . 1 . . 1 1 1 1 1 1 1 1 . . 1 . . 1 1 . 1 . 1 . 1 1
. . 1 . 1 1 . 1 . . 1 . 1 . . . 1 . 1 . . . . 1 . . 1 1 . . . 1
1 1 . . . . . 1 1 1 1 1 1 . 1 1 . 1 . 1 . . 1 . 1 1 . . 1 . . 1
1 . . 1 1 1 1 1 1 . 1 1 1 . 1 . . . 1 . 1 1 . 1 . 1 . . 1 1 1 1
1 . . 1 1 . 1 . . . 1 1 1 . . . . 1 1 . . . . 1 . . 1 . . 1 1 .
. . . 1 . 1 1 . . 1 . . . 1 1 . 1 . . 1 . 1 . . . . 1 1 1 1 1 1
. . 1 . 1 1 1 . 1 . . . . 1 . 1 1 . . . . . 1 1 1 1 1 1 1 1 1 .
1 1 . . . . . 1 1 . . 1 . . 1 . . 1 . 1 1 . 1 . . 1 1 1 . 1 . .
answered Mar 11, 2015 at 10:48
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2$\begingroup$ Quite a comprehensive answer! I just did mine my hand experiment- it didn't take long in this cae. $\endgroup$ Commented Mar 11, 2015 at 10:51