Eigenvalue Equation
Defines eigenvalues and eigenvectors of a matrix.
Formula
Variables
Example
For A = [[2,0],[0,3]], v = (1,0) is an eigenvector with eigenvalue λ = 2.
How it works
The equation A\vec{v} = \lambda\vec{v} says that after a matrix acts on a nonzero vector, the vector does not change direction: it is only stretched, shrunk, or reversed. That special vector \vec{v} is an eigenvector, and the scaling factor \lambda is its eigenvalue. It works because we are looking for vectors that lie on a one-dimensional invariant subspace of the linear transformation, so the transformation acts like multiplication by a scalar along that direction.
When to use it
Use this when you want to find directions that a linear transformation preserves, such as principal axes in geometry, normal modes in vibrations, or steady-state behavior in dynamical systems. It is also central in diagonalization, stability analysis, PCA, and solving differential equations with linear systems.
Worked example, step by step
Take A = [[4,1],[0,2]]. Let \vec{v} = (1,0). Compute A\vec{v} = [[4,1],[0,2]](1,0) = (4,0). Now compare (4,0) with \lambda(1,0). We need \lambda(1,0) = (4,0), so \lambda = 4. Therefore, \vec{v} = (1,0) is an eigenvector of A with eigenvalue \lambda = 4. Check: A\vec{v} = (4,0) = 4(1,0) = \lambda\vec{v}.
Common mistakes to avoid
- Forgetting that the eigenvector must be nonzero
- Assuming every vector is an eigenvector instead of only special directions
- Mixing up the eigenvalue equation with the characteristic equation det(A - \lambda I) = 0
Did You Know?
Eigenvectors are the special directions a matrix merely stretches — the math behind Google's original PageRank.
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