In connection with tensors, matrices are of interest for two reasons. Firstly, they are tensors of order two and therefore a nontrivial example of a tensor. Differently from tensors of higher order, matrices allow to apply practically realisable decompositions. Secondly, operations with general tensors will often be reduced to a sequence of matrix operations (realised by well-developed software).
introduce the notation and recall well-known facts about matrices.
discusses the important QR decomposition and the singular value decomposition (SVD) and their computational cost. The (optimal) approximation by matrices of lower rank explained in
will be used later in truncation procedures for tensors. In Part III we shall apply some linear algebra procedures introduced in
based on QR and SVD.