The problem of matrix eigenvalues is encountered in various fields of engineering
endeavor. In this paper, a new approach based on the Adomian decomposition method and the
Faddeev-Leverrier's algorithm is presented for finding real eigenvalues of any desired real matrices.
The method features accuracy and simplicity. In contrast to many previous techniques which merely
afford one specific eigenvalue of a matrix, the method has the potential to provide all real
eigenvalues. Also, the method does not require any initial guesses in its starting point unlike most
of iterative techniques. For the sake of illustration, several numerical examples are included.