### Abstract:

Chapter 1 reviews better-known factorization theorems of a square
matrix. For example, a square matrix over a field can be expressed
as a product of two symmetric matrices; thus square matrices over
real numbers can be factorized into two diagonalizable matrices.
Factorizing matrices over complex num hers into Hermitian matrices
is discussed. The chapter concludes with theorems that enable one to
prescribe the eigenvalues of the factors of a square matrix, with
some degree of freedom. Chapter 2 proves that a square matrix over
arbitrary fields (with one exception) can be expressed as a product
of two diagona lizab le matrices. The next two chapters consider
decomposition of singular matrices into Idempotent matrices, and of
nonsingutar matrices into Involutions. Chapter 5 studies
factorization of a comp 1 ex matrix into Positive-( semi )definite
matrices, emphasizing the least number of such factors required