Applications of full rank factorization to solving matrix equations

In the study of matrices, we are always searching for tools which allow us to simplify our investigations. Because full rank factorizations exist for all matrices and their properties often help to simplify arguments, their uses are abundant. There exist many matrix equations for which solutions are otherwise quite difficult to find. Full rank factorizations and generalized inverses allow us to easily find solutions to many such equations. Their properties can also be used to study the diagonalization of non-square matrices and to develop conditions under which matrices are simultaneously diagonalizable. Finally, the full rank factorization can be used to derive canonical forms and other factorizations such as the singular value decomposition.