Find directions a transformation preserves and scales
An eigenvector is a nonzero vector whose direction remains on the same line after transformation. Its eigenvalue records the scale factor, including sign. Eigenanalysis reveals natural modes, long-term dynamics, stability, vibration shapes, Markov behavior, and principal directions of symmetric operators.
The eigenvalue equation
Turn a nonlinear-looking search into a singularity condition
A nonzero solution exists only when A−λI has a nontrivial null space, which occurs precisely when det(A−λI)=0. This determinant is the characteristic polynomial.
The zero vector is never an eigenvector
Although A0=λ0 for every λ, the zero vector carries no direction and is explicitly excluded.
Eigenspaces and multiplicity
Distinguish repeated roots from independent eigenvector directions
Algebraic multiplicity
The number of times λ appears as a root of the characteristic polynomial.
Geometric multiplicity
dim N(A−λI), the number of independent eigenvector directions for λ.
Geometric multiplicity is at least one and no larger than algebraic multiplicity. A matrix is diagonalizable when it has enough independent eigenvectors to form a basis.
Diagonalization and repeated application
Replace a difficult matrix power with scalar powers
The columns of P are an eigenvector basis and D contains corresponding eigenvalues on its diagonal. In that basis, the transformation acts by independent coordinate scaling. A defective matrix lacks enough eigenvectors and requires more general structures such as Jordan form.
Spectral theorem for real symmetric matrices
Every real symmetric matrix has real eigenvalues and an orthonormal eigenbasis. It can be diagonalized as A=QΛQᵀ with Q orthogonal.
Interactive eigenanalysis laboratory
Solve 2×2 real or complex eigenproblems and inspect power iteration
2×2 eigenvalue and eigenvector analyzer
The solver uses trace, determinant, and the characteristic discriminant.
Power-iteration trace
Applications and stability
Interpret eigenvalues in context rather than as isolated roots
| Context | Eigenvalue meaning | Typical interpretation |
|---|---|---|
| Discrete dynamics xₖ₊₁=Axₖ | Growth factor per step | |λ|<1 decays, |λ|>1 grows, negative λ alternates direction. |
| Differential equations x′=Ax | Exponential rate e^{λt} | Negative real part decays; positive real part grows. |
| Vibrations | Natural mode parameter | Eigenvectors are mode shapes. |
| Markov chains | Long-term modes | λ=1 often corresponds to a stationary distribution. |
| PCA / covariance | Variance along a principal direction | Largest eigenvalues identify dominant data directions. |
Review questions
Test eigenvalue equations, multiplicity, and diagonalization