This graduate thesis is dedicated to the eigenvalues of the matrix representation of linear operators. The thesis is divided into four chapters. The first chapter contains a reminder of basic concepts such as the concept of linear operator and square matrix and the connection between these two concepts. In the same chapter, the definitions of the eigenvalues of the linear operator and its spectrum are given, as well as examples of the same, of which the mirroring operators stand out in particular. The procedure for finding eigenvalues is also described, and the theorem for the necessary and sufficient condition that a scalar is the desired eigenvalue is stated and proved. There is also an example of an operator that has no eigenvalues. The second chapter is devoted to the necessary and sufficient conditions for the diagonalization of operators. We list the definitions of eigen subspace, geometric and algebraic multiplicity. A diagonal matrix is a practical form of matrix from which we can determine many useful information, but sometimes this cannot be done. Part of the second chapter is dedicated to such matrices. We state and describe the procedure for determining an elegant form of the matrix, which we call the Jordan form of the matrix. In the third chapter, we deal with determining the approximate location of the eigenvalues of linear operators when it’s more difficult to find them using the procedure described in the previous chapters. We define Geršgorin’s circles and state and prove Geršgorin’s theorem, which is also interpreted with concrete examples. Eigenvalues and eigenvectors are widely used in various physical fields such as quantum mechanics, where they are often used to describe wave functions and spin operators. Very often, eigenvectors describe some types of particle charges (in the physical sense). In the last, fourth chapter, we present an interesting application of eigenvalues on an Internet search engine. We are forming a ”little internet” where we show how the so-called PageRank algorithm works, which is the basis of the most famous internet search engine, Google. Accordingly, we construct a matrix that we call the Google matrix and describe its properties.