One way to intuitively describe a matrix in 3D is to represent it in terms of its eigenvectors and eigenvalues. Depending on whether the eigenvalues are real or complex, all different from each other or not, the eigenvectors build up either three characteristic directions, or one surface of rotational dynamics additional to one characteristic direction.
Hyper stream lines by Delmarcelle and
Hesselink  are a visualization concept for
tensor data based on the decomposition described above.
Certain characteristic curves are integrated
like stream lines for flow data. Such curves follow, for
example, the direction of maximum stress propagation.
See Fig. 2.11 of a typical image in this area.