Note on the factorization of a square matrix into two hermitian or symmetric matrices

Introduction: Although the results already have been published (partially) by Frobenius in 1910 (see [5]), these are still not very known to mathematicians. I even could not find them in modern textbooks on matrix theory or linear algebra. These results and their proofs (see [1] , [2], [3]) are not very accessible for non-mathematicians. But they need the results. Applications can be found in system theory and in problems in mechanics concerning systems of differential equations. The aim of this paper is to give elementary proofs as well as a clear summary of the conditions. The basis of all proofs is the Jordan normal form. As we will see: every square matrix (real or complex) is a product of two symmetric (real resp. complex) matrices. However, not every complex square matrix is a product of two hermitian matrices.