By matrix decomposition, here it means a factorization of a matrix into a product of two or more simpler matrices. Thus, the word “decomposition” here is interchangeable with the word “factorization.” In some other situations, matrix decomposition may include writing a given matrix into a sum of two or more matrices.

There are many types of matrix decompositions. Some of them are useful in revealing the structure and properties of a given matrix, while some are useful in developing algorithms for computations. In this entry, we introduce some basic decompositions for complex matrices. Square matrices are considered to simplify our discussion. In some situations, modifications are needed if real matrices are involved, especially when a decomposition gives the eigenvalues, but the real matrix under consideration has complex eigenvalues. Our general references include Golub and Van Loan (2013), Horn and Johnson (2012), Jin and Wei (2012), and Stewart (1998).

The decompositions discussed in this entry are divided into the following four categories:

1.Decompositions involving (upper or lower) triangular matrices

2.Classical decompositions under similarity

3.Spectral and singular value decompositions

4.Decompositions of some particular types of matrices