Motivic Splittings For Symmetric Matrices

We show that the space of symmetric matrices of a fixed rank $k$ over a field $K$ of characteristic not equal to $2$ is split Tate. We do this by promoting the point-counting strategy of MacWilliams over finite fields to a filtration of the locus of rank $\leq k$ symmetric matrices that is independent of the field. This filtration immediately allows for a computation of their isomorphism classes in the Grothendieck ring of varieties in terms of the Lefschetz motive. We then promote this computation to prove that the space of symmetric matrices of a fixed rank $k$ are split Tate in Voevodsky's category of motives in characteristic $0$ and Kelly's category of motives in characteristic $p$.