If $A$ is a symmetric integral matrix with zero diagonal, then I want to prove $2-rank(A)$ (i.e. the dimension of $C_A$ ) is even?
$2-rank(A)$ means dimension A on field $\mathbb{F}_2$.
If $A$ is a symmetric integral matrix with zero diagonal, then I want to prove $2-rank(A)$ (i.e. the dimension of $C_A$ ) is even?
$2-rank(A)$ means dimension A on field $\mathbb{F}_2$.
When considering elements of $\mathbb{F}_2$ a symmetric matrix with zero diagonal values is actually a skew symmetric matrix since $1 \equiv -1 \mod(2)$. It is easier to prove that a (real) skew symmetric matrix of odd dimension has zero determinant.