  In this paper, we introduce a Helmholtz-type decomposition for the space of
square integrable, symmetric-matrix-valued functions analogous to the standard
Helmholtz decomposition for vector fields. This decomposition provides a better
understanding of the strain constraint space, which is important to the
Navier--Stokes regularity problem. In particular, we give a full
characterization the orthogonal complement of the strain constraint space and
investigate the geometry of the eigenvalue distribution of matrices in the
strain constraint space.
