Iterative Algorithms for Eigenvalues
Numerical calculation of eigenvalues and eigenvectors uses a different approach from classical methods. Traditional methods typically find zeros of the characteristic polynomial, but in numerical computation we use iterative algorithms that gradually approach eigenvalues and eigenvectors.
Imagine an archer throwing arrows. Instead of hitting the center of the target in one shot (like finding polynomial roots directly), the archer uses repeated practice techniques to get closer and closer to the target (like iterative algorithms). Each throw makes them more accurate.
Nonlinear Problems
Eigenvalue calculation problems fall into the category of nonlinear problems. As a result, direct methods from linear algebra such as matrix decomposition alone do not provide adequate solutions. We need special approaches that can handle this nonlinear complexity.