Eigenvalue
An eigenvalue of a square matrix A is a scalar λ such that there exists a non-zero vector v (its eigenvector) satisfying Av = λv, meaning the matrix only scales the vector without changing its direction. Eigenvalues are found by solving the characteristic equation det(A − λI) = 0, and they encode fundamental properties of the linear transformation including stability, principal stresses, and oscillation frequencies. Applications span quantum mechanics (energy levels), structural engineering (natural frequencies), principal component analysis (data variance), and PageRank (web link importance).
Key Formula
det(A - lambda * I) = 0
LaTeX: \det(A - \lambda I) = 0
| Symbol | Meaning | Unit |
|---|---|---|
| A | Square matrix | — |
| \lambda | Eigenvalue (scalar) | — |
| I | Identity matrix of same size as A | — |
Worked Example
Problem
Find the eigenvalues of A = [[4, 1], [2, 3]].
Solution
Step 1: Form the characteristic equation: det(A − λI) = 0. Step 2: A − λI = [[4−λ, 1], [2, 3−λ]]. Step 3: det = (4−λ)(3−λ) − (1)(2) = 12 − 7λ + λ² − 2 = λ² − 7λ + 10. Step 4: Solve λ² − 7λ + 10 = 0: (λ − 5)(λ − 2) = 0.
Answer
Eigenvalues are λ₁ = 5 and λ₂ = 2.
Eigenvalues in Different Applications
| Field | Matrix Represents | Eigenvalue Meaning |
|---|---|---|
| Quantum mechanics | Hamiltonian operator | Energy level of quantum state |
| Structural engineering | Stiffness matrix | Natural vibration frequency (squared) |
| Statistics / PCA | Covariance matrix | Variance along principal component |
| Graph theory | Adjacency matrix | Spread / connectivity of graph |
| Differential equations | System matrix | Rate of exponential growth/decay |
Interactive Tools
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Related Terms
Eigenvector
An eigenvector of a square matrix A is a non-zero vector v that, when multiplied by A, yields a scalar multiple of itself: Av = λv, where λ is the corresponding eigenvalue. Geometrically, eigenvectors are the special directions in space that a linear transformation leaves invariant — only stretching or compressing along those directions without rotation. Eigenvectors are the foundation of matrix diagonalisation, principal component analysis, Google's PageRank algorithm, and the description of quantum states in physics.
Determinant
The determinant is a scalar value computed from a square matrix that encodes whether the linear transformation represented by the matrix is invertible, scaling the volume by a factor equal to the absolute value of the determinant and changing orientation if the determinant is negative. For a 2×2 matrix, det(A) = ad − bc; for larger matrices, it is computed recursively via cofactor expansion or row reduction. The determinant is zero if and only if the matrix is singular (non-invertible), making it a critical tool in solving linear systems, computing eigenvalues, and computing cross products in geometry.
Matrix (Linear Algebra)
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns, where an m×n matrix has m rows and n columns. Matrices represent linear transformations between vector spaces, encode systems of linear equations, and serve as the primary computational tool in linear algebra. They are indispensable across engineering, physics, computer science, statistics, and machine learning, where they model data, rotations, reflections, and the weights of neural networks.
From German "eigen" (own, characteristic) + "Wert" (value), meaning "characteristic value". The term was popularised by David Hilbert around 1904; earlier related concepts appear in Cauchy's work on quadratic forms (1829).