Publisher Summary This chapter reviews the matrix theory and linear differential equations as well as the application of matrix theory to linear differential equations. The chapter discusses the vector-matrix notation. A column of numbers is called a “vector.” Two vectors x and y are equal if their respective components are equal. A square array of numbers is called a “matrix.” Matrix multiplication is noncommutative. At first, this violation of the most important property of the ordinary multiplication appears to be a terrible nuisance. A little reflection will convince the reader that it is actually an enormous boon. Many important transformations in the scientific domain are noncommutative and the matrix theory enables to obtain a mathematical foothold. The chapter shows that orthogonal matrices form a group, that is, that the product of two orthogonal matrices is again orthogonal, that every orthogonal matrix has an inverse that is orthogonal, and that the identity matrix is an orthogonal matrix.