ABSTRACT
This chapter explains some important topics from linear algebra that plays an essential role in the study of systems of differential equations. It describes four methods: diagonalization, Sylvester's, resolvent, and the spectral decomposition procedure, of which the last three do not require any knowledge about eigenvectors. The chapter shows the reduction of matrices into their diagonal form. This approach, called diagonalization, allows to define a function of a square matrix. The chapter presents the diagonalization procedure only for matrices with nondefective eigenvalues, that is, when the algebraic and geometric multiplicities are equal for every eigenvalue. The algorithm based on diagonalization can be extended for an arbitrary matrix; however, all calculations become more tedious as a result and require more work. The chapter provides probably the most effective algorithm to define a function of a square matrix—the Sylvester method.
