What Are Eigenvectors?
An eigenvector of a square matrix A is a non-zero vector v that, when multiplied by A, results in a scalar multiple of itself: Av = λv. The scalar λ is the corresponding eigenvalue. In geometric terms, eigenvectors are the special directions along which a linear transformation acts as simple scaling — the vector's direction is preserved (or reversed if λ is negative), and only its magnitude changes by the factor λ.
Eigenvectors are central to linear algebra and appear throughout applied mathematics. In principal component analysis (PCA), eigenvectors of the covariance matrix define the principal components — the directions of maximum variance in a dataset. In differential equations, eigenvectors determine the fundamental solution modes. In quantum mechanics, eigenvectors represent the observable states of a physical system.
How to Find Eigenvectors
Finding eigenvectors is a two-step process. First, find the eigenvalues by solving the characteristic equation det(A - λI) = 0. Second, for each eigenvalue λ, solve the system (A - λI)v = 0 to find the corresponding eigenvector(s). This system is always homogeneous (right-hand side is zero), so the eigenvector is determined up to a scalar multiple — any non-zero scalar multiple of an eigenvector is also an eigenvector.
For a 2×2 matrix [[a, b], [c, d]] with eigenvalue λ, the system (A - λI)v = 0 becomes: (a-λ)v₁ + bv₂ = 0 and cv₁ + (d-λ)v₂ = 0. Since the system is homogeneous with a singular coefficient matrix, we can solve either equation for the ratio v₁/v₂. A common convention is to set v₁ = 1 and compute v₂, or to normalize the eigenvector to unit length.
Worked Example
Consider A = [[3, 1], [0, 2]]. The characteristic equation is (3-λ)(2-λ) = 0, giving eigenvalues λ₁ = 3 and λ₂ = 2.
For λ₁ = 3: (A - 3I)v = [[0, 1], [0, -1]]v = 0. From the first row: v₂ = 0. So v₁ is free (any value), giving eigenvector v₁ = (1, 0).
For λ₂ = 2: (A - 2I)v = [[1, 1], [0, 0]]v = 0. From the first row: v₁ + v₂ = 0, so v₂ = -v₁. Setting v₁ = 1 gives eigenvector v₂ = (1, -1).
Special Cases
Repeated eigenvalues: When a matrix has a repeated eigenvalue, it may have one or two linearly independent eigenvectors. If it has only one, the matrix is called defective. A 2×2 matrix with a repeated eigenvalue has two independent eigenvectors only if it is a scalar multiple of the identity matrix.
Complex eigenvalues: When a real matrix has complex eigenvalues (which always come in conjugate pairs), the eigenvectors are also complex. The real and imaginary parts of a complex eigenvector provide the directions of rotation and scaling that characterize the transformation.
Symmetric matrices: Real symmetric matrices always have real eigenvalues and orthogonal eigenvectors. This is the spectral theorem, and it makes symmetric matrices especially well-behaved for diagonalization and principal component analysis.
Applications of Eigenvectors
Diagonalization: A matrix A with n linearly independent eigenvectors can be decomposed as A = PDP⁻¹, where P is the matrix of eigenvectors and D is the diagonal matrix of eigenvalues. This makes computing powers Aⁿ trivial: Aⁿ = PDⁿP⁻¹.
Google PageRank: The ranking of web pages is determined by the dominant eigenvector of the web link matrix. Pages with higher eigenvector components are ranked higher in search results.
Vibration analysis: In structural engineering, eigenvectors of the mass-stiffness matrix represent the mode shapes — the patterns of vibration at each natural frequency. Understanding these modes is critical for designing structures that resist resonance.
Image compression: Eigenvectors enable techniques like eigenfaces in facial recognition, where a set of images is decomposed into principal eigenvectors that capture the most important variations in appearance.