Products of diagonalizable matrices

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 numbers 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 diagonalizable matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingular matrices into Involutions. Chapter 5 studies factorization of a complex matrix into Positive-(semi)definite matrices, emphasizing the least number of such factors required.