  In 1954, Alston S. Householder published \textit{Principles of Numerical
Analysis}, one of the first modern treatments on matrix decomposition that
favored a (block) LU decomposition-the factorization of a matrix into the
product of lower and upper triangular matrices. And now, matrix decomposition
has become a core technology in machine learning, largely due to the
development of the back propagation algorithm in fitting a neural network. The
sole aim of this survey is to give a self-contained introduction to concepts
and mathematical tools in numerical linear algebra and matrix analysis in order
to seamlessly introduce matrix decomposition techniques and their applications
in subsequent sections. However, we clearly realize our inability to cover all
the useful and interesting results concerning matrix decomposition and given
the paucity of scope to present this discussion, e.g., the separated analysis
of the Euclidean space, Hermitian space, Hilbert space, and things in the
complex domain. We refer the reader to literature in the field of linear
algebra for a more detailed introduction to the related fields.
  Keywords: Existence and computing of matrix decompositions, Floating point
operations (flops), Low-rank approximation, Pivot, LU/PLU decomposition,
CR/CUR/Skeleton decomposition, Coordinate transformation, ULV/URV
decomposition, Rank decomposition, Rank revealing decomposition,
Update/downdate, Tensor decomposition.
