  We discuss extended definitions of linear and multilinear operations such as
Kronecker, Hadamard, and contracted products, and establish links between them
for tensor calculus. Then we introduce effective low-rank tensor approximation
techniques including Candecomp/Parafac (CP), Tucker, and tensor train (TT)
decompositions with a number of mathematical and graphical representations. We
also provide a brief review of mathematical properties of the TT decomposition
as a low-rank approximation technique. With the aim of breaking the
curse-of-dimensionality in large-scale numerical analysis, we describe basic
operations on large-scale vectors, matrices, and high-order tensors represented
by TT decomposition. The proposed representations can be used for describing
numerical methods based on TT decomposition for solving large-scale
optimization problems such as systems of linear equations and symmetric
eigenvalue problems.
