The Pi (or Pie) Plane showing the region of the principal solution for the Lode angle and hence the region nearest the max eigenvalue.

It probably isn’t surprising that an exact solution can be found for eigenvalues of a real 3×3 symmetric matrix.  This conclusion follows from noting that the characteristic equation is cubic, for which an exact solution procedure can be found in any good algebra reference.  One thing that you won’t find in many such resources, however, is an algorithm for the solution that will avoid complex numbers in the intermediate steps of the calculation whenever the components of the source symmetric 3×3 matrix are all real.

Having a computational algorithm guaranteed to NOT involve any complex numbers as intermediate steps is valuable for computer implementation. Moreover, having the exact algebraic solutions for the eigenvalues is very useful to then be able to differentiate the eigenvalues with respect to the source matrix components.

Another thing that is even more rarely seen in books is an exact solution for the eigen-projectors (whose columns may be orthonormalized, if desired, to obtain eigenvectors).  Brannon’s as-yet-unpublished book on tensor analysis covers this topic. Just click here for the relevant excerpt!.   Warning: the execerpt has a typo: please replace the formula for the low eigenvalue with the one shown in the summary box below (only difference is the sign of the term in brackets).

For a working Mathematica file, click here.