  The spectral decomposition of a symmetric, second-order tensor is widely
adopted in many fields of Computational Mechanics. As an example, in
elasto-plasticity under large strain and rotations, given the Cauchy
deformation tensor, it is a fundamental step to compute the logarithmic strain
tensor.
  Recently, this approach has been also adopted in small-strain isotropic
plasticity to reconstruct the stress tensor as a function of its eigenvalues,
allowing the formulation of predictor-corrector return algorithms in the
invariants space. These algorithms not only reduce the number of unknowns at
the constitutive level, but also allow the correct handling of stress states in
which the plastic normals are undefined, thus ensuring a better convergence
with respect to the standard approach.
  While the eigenvalues of a symmetric, second-order tensor can be simply
computed as a function of the tensor invariants, the computation of its
eigenbasis can be more difficult, especially when two or more eigenvalues are
coincident. Moreover, when a Newton-Rhapson algorithm is adopted to solve
nonlinear problems in Computational Mechanics, also the tensorial derivatives
of the eigenbasis, whose computation is still more complicate, are required to
assemble the tangent matrix.
  A simple and comprehensive method is presented, which can be adopted to
compute a closed form representation of a second-order tensor, as well as their
derivatives with respect to the tensor itself, allowing a simpler
implementation of spectral decomposition of a tensor in Computational Mechanics
applications.
