Matrix factorization has a great significance in matrix theory, as well as in Numerical Linear Algebra. Schur decomposition is considered as one of the basic tools, when it comes to analyzes and finding a numerical solution to an eigenvalue problem. This decomposition in particular is proven to give an answer to the question, to which level a matrix \( A ∈ C^{ n×n}\) . can be simplified by the use of unitary transformations of similarity. Eigenvalues and eigenvectors of matrix A are transferred to the matrici \(B = S^{-1}AS\), which is similar to the matrix A. So, if matrices A and B are similar, following equation Bx = λx implies that ASx = λSx. This process implies that it can be practical to use a matrix B, which has a simplified structure in order to calculate the eigenvalues and eigenvectors of matrix A. The process of unitary transformation of similarity which transforms a matrix A to an upper triangular matrix is given by the Schur decomposition. Unitary transformation of similarity S is proven to have some advantages when it comes to its theoretical and numerical properties. This particular case allows as to calculate the inverse of matrix of transformation more easily, as \(S^{-1}=S^{∗}\) , and the norm of perturbations is not enlarged during the process, as a class of matrix is preserved. For instance, if matrix A is normal, hermitian, antihermitian or unitary, than matrix \(B = S^{∗}AS)\) is also normal, hermitian, antihermitian or unitary, respectively. In this paper the results of Schur decomposition are represented. The process of Schur decomposition (real and complex) is based on the use of the Gram-Schmidt orthogonalization process. Examples represented in this paper have covered both real and complex cases, so that the process can be fully illustrated. Significance of Schur theorem is stated in terms of well known theorems which are all results of Schur decomposition process