Could someone please give me the simpleton explanation to eigenvalues and vectors. What is their purpose? What are the various parts of basic formula A*X=X*r (with A and X as matrices, X the eigenvector and r the eigenvalue? How does this relate to homogeneous linear system?

Question

Could someone please give me the simpleton explanation to eigenvalues and vectors.  What is their purpose?  What are the various parts of basic formula A*X=X*r (with A and X as matrices, X the eigenvector and r the eigenvalue? How does this relate to homogeneous linear system?

ybarrap · Answer

Eigenvectors decompose the solutions to linearly independent vectors. The eigenvalues determine if if solutions decay or grow. The eigenvalues are related to homogeneous linear systems in that the roots of these systems correspond to these eigenvalues.