  We focus on symmetries related to matrices and vectors appearing in the
simulation of quantum many-body systems. Spin Hamiltonians have special
matrix-symmetry properties such as persymmetry. Furthermore, the systems may
exhibit physical symmetries translating into symmetry properties of the
eigenvectors of interest. Both types of symmetry can be exploited in sparse
representation formats such as Matrix Product States (MPS) for the desired
eigenvectors.
  This paper summarizes symmetries of Hamiltonians for typical physical systems
such as the Ising model and lists resulting properties of the related
eigenvectors. Based on an overview of Matrix Product States (Tensor Trains or
Tensor Chains) and their canonical normal forms we show how symmetry properties
of the vector translate into relations between the MPS matrices and, in turn,
which symmetry properties result from relations within the MPS matrices. In
this context we analyze different kinds of symmetries and derive appropriate
normal forms for MPS representing these symmetries. Exploiting such symmetries
by using these normal forms will lead to a reduction in the number of degrees
of freedom in the MPS matrices. This paper provides a uniform platform for both
well-known and new results which are presented from the (multi-)linear algebra
point of view.
