Eigenvalue and Eigenvector Formula
How to find eigenvalues and eigenvectors of a matrix.
Fundamental to linear algebra with applications in PCA, quantum mechanics, and vibration analysis.
The Formula
det(A − λI) = 0 (characteristic equation)
For a square matrix A, an eigenvector v is a nonzero vector that only changes by a scalar factor when A is applied to it. That scalar factor is the eigenvalue λ. To find λ, solve the characteristic equation. To find v for each λ, solve the homogeneous system (A − λI)v = 0.
Variables
| Symbol | Meaning |
|---|---|
| A | Square n × n matrix |
| v | Eigenvector (nonzero column vector) |
| λ | Eigenvalue (scalar) |
| I | Identity matrix (same size as A) |
| det() | Determinant |
An n × n matrix has exactly n eigenvalues (counting multiplicity), which may be real or complex. The characteristic polynomial det(A − λI) is a polynomial of degree n in λ.
Example — 2×2 Matrix
Find the eigenvalues and eigenvectors of A = [[4, 1], [2, 3]]
Characteristic equation: det(A − λI) = 0
det([[4−λ, 1], [2, 3−λ]]) = (4−λ)(3−λ) − 2 = 0
λ² − 7λ + 12 − 2 = λ² − 7λ + 10 = 0
(λ − 5)(λ − 2) = 0 → λ&sub1; = 5, λ&sub2; = 2
For λ&sub1; = 5: (A − 5I)v = 0 → [[-1, 1],[2,-2]]v = 0 → v = [1, 1]
For λ&sub2; = 2: (A − 2I)v = 0 → [[2,1],[2,1]]v = 0 → v = [1, −2]
When to Use It
Eigenvalues and eigenvectors appear throughout science and engineering:
- Principal Component Analysis (PCA): eigenvectors of the covariance matrix define the principal components; eigenvalues give variance explained
- Quantum mechanics: energy levels are eigenvalues of the Hamiltonian operator; wave functions are eigenstates
- Structural engineering: natural vibration frequencies are eigenvalues; mode shapes are eigenvectors
- Differential equations: solving systems y' = Ay uses eigendecomposition
- Google's PageRank: web page importance is the dominant eigenvector of a link matrix
- Image compression: Singular Value Decomposition (SVD) extends eigenanalysis to rectangular matrices
A symmetric matrix always has real eigenvalues and orthogonal eigenvectors. A diagonal matrix's eigenvalues are simply its diagonal entries, with standard basis vectors as eigenvectors.