1. Get to know: Eigenvalue & Eigenvector 2. Estimation of Eigenvalue & Eigenvector 3. Theorem
derived from the German word "eigen“ means "proper" or "characteristic." Eigenvalues and the associated eigenvectors ‘special’ properties of square matrices (n x n) Eigenvalues parameterize the dynamical properties of the system (timescales, resonance properties, amplification factors, etc) Eigenvectors define the vector coordinates of the normal modes of the system.
A: a Linear Transformation Square Matrix (n x n) x: Eigenvector (non-zero vector) of A (not unique) ג : Eigenvalue (Scalar value) of A
The general state of the system can be expressed as a linear combination of eigenvectors.
The beauty of eigenvectors is that They can be made orthogonal (decoupled from one another). An orthogonal expansion of the system is possible. The normal modes can be handled independently
The dominant eigenvector of a matrix A an eigenvector corresponding to the eigenvalue of largest magnitude (for real numbers, largest absolute value) of that matrix. Many of the "real world" applications are primarily interested in the dominant eigenpair. The method used to find the dominant eigenvector is called the power method.
The eigenvalue of smallest magnitude of a matrix is the same as the inverse (reciprocal) of the dominant eigenvalue of the inverse of the matrix. If the eigenvalue of smallest magnitude is needed, the inverse matrix A is often used to solve for its dominant eigenvalue. This is why the dominant eigenvalue is so important.