Concept
Vectors a matrix only stretches without rotating. The "principal axes" of high-dimensional data — PageRank, PCA, quantum mechanics.
Understand it in one breath
A nonzero vector v that remains on the same line after multiplication by A, scaled by λ. A negative λ reverses orientation, and complex eigenvalues may be needed. PageRank models links as a stochastic transition matrix and approximates an eigenvector corresponding to its stationary distribution.
At a glance
Matrix A | Eigenvalue λ | Eigenvector v | Intuition |
|---|---|---|---|
[[2,0],[0,3]] | 2, 3 | (1,0), (0,1) | Only the x- and y-axis directions stay fixed |
[[3,1],[0,2]] | 3, 2 | (1,0), (1,−1) | Two tilted invariant axes |
90° rotation [[0,-1],[1,0]] | Complex ±i | (1, ±i) | No real eigenvectors |
PCA data covariance | Principal-component variance | Principal-axis direction | The data’s dominant directions |
Google link matrix | 1 (scalar) | PageRank vector | Stationary distribution of the web |
Av = λv. Over the complex numbers, the characteristic polynomial of an n×n matrix has n roots counting multiplicity, but the matrix need not have n linearly independent eigenvectors.
Key formula
Av = λv
Modern applications
PCA dimensionality reduction, PageRank, energy states in quantum mechanics, and vibration-mode analysis.
Beyond MathVoyage
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