Eigenvalues and Eigenvectors

Special directions a matrix only stretches, and the factors by which it stretches them.

The defining equation

A nonzero vector v is an eigenvector of a square matrix A if A v = lambda v for some scalar lambda, the corresponding eigenvalue. In words, A acts on v purely by scaling, without changing its direction. Eigenvectors reveal the natural axes of a transformation, along which its action is simplest.

Finding them

Rearranging gives (A - lambda I) v = 0, which has a nonzero solution only when A - lambda I is singular, that is when det(A - lambda I) = 0. This is the characteristic equation, a polynomial of degree n whose roots are the eigenvalues. For each eigenvalue, the null space of A - lambda I gives the eigenvectors.

What eigenvalues tell you

• their product equals the determinant
• their sum equals the trace
• a zero eigenvalue means the matrix is singular
• for symmetric matrices they are all real

Why they matter

Eigenvalues govern stability and long-term behavior. In a discrete dynamical system x_{k+1} = A x_k, the state grows if any eigenvalue exceeds one in magnitude and decays if all are below one. In differential equations dx/dt = A x, the sign of the real parts decides stability. This makes eigenvalue analysis central to control, vibration, and plasma stability.

import numpy as np
A = np.array([[2.0, 1.0], [1.0, 2.0]])
vals, vecs = np.linalg.eig(A)
print(vals)         # [3. 1.]
print(vecs)         # columns are eigenvectors

Stability of a magnetized plasma is analyzed by linearizing the governing equations and examining the eigenvalues of the resulting operator; growth-rate eigenvalues flag instabilities that a confinement design such as Hyperion must avoid.