Computation of Eigenvalues

Vector $$\textbf{x}$$ x is an eigenvector of a matrix $$\textbf{A}$$ A if the following equation is satisfied: $$ \textbf{Ax}=\lambda \textbf{x}$$ Ax = λ x . Scalar $$\lambda $$ λ is known as eigenvalue of the eigenvector. System of equations which can be reduced to the above form are classified as eigenvalue problems. One might wonder what the importance of eigenvalues in practical engineering problems is. As such, there are several practical examples which can be reduced to eigenvalue problems. Eigenvalues form an important foundation for quantum mechanics. Eigenvalues are employed in data analysis tools such as principal component analysis. The most remarkable example of eigenvalue used for data analysis is the algorithm behind Google’s search algorithm. Before treating the eigenvalues mathematically, we try to understand what eigenvalues represent in physical systems.