Edit the 2×2 matrix A and observe how it transforms the plane. The unit circle maps to an ellipse; eigenvectors mark the principal axes — directions that only scale without rotating.
Characteristic: λ² - tr(A)λ + det(A) = 0
λ₁,₂ = (tr(A) ± √(tr²-4det)) / 2
Eigenvector: (A - λI)v = 0
|A| = λ₁·λ₂, tr(A) = λ₁+λ₂
Google's original PageRank algorithm finds the dominant eigenvector of the web's link matrix — the page with the largest eigenvalue component ranks highest. Eigendecomposition underpins quantum mechanics, PCA, vibration analysis, and facial recognition.
What is an eigenvalue?
An eigenvalue λ of matrix A is a scalar where Av = λv for a non-zero vector v. It measures how much the transformation stretches or compresses space along the eigenvector direction.
What is an eigenvector?
An eigenvector is a non-zero vector whose direction is unchanged by the matrix transformation — it only gets scaled by its eigenvalue. Eigenvectors point along the principal axes of the transformation.
How do you find eigenvalues of a 2×2 matrix?
Solve det(A − λI) = 0, which gives λ² − tr(A)·λ + det(A) = 0. Use the quadratic formula: λ = [tr(A) ± √(tr(A)² − 4·det(A))] / 2.
The unit circle maps to an ellipse. For symmetric matrices, the ellipse's semi-axes align exactly with the eigenvectors, and their lengths equal the absolute eigenvalues, making eigenvectors the true principal axes.
Complex eigenvalues indicate a rotational component. No real eigenvectors exist. Points spiral inward if |λ| < 1, outward if |λ| > 1, or orbit on ellipses if |λ| = 1. Rotation matrices have purely imaginary eigenvalues of modulus 1.
p(λ) = det(A − λI). For 2×2 matrices: p(λ) = λ² − tr(A)·λ + det(A). By the Cayley-Hamilton theorem, every matrix satisfies its own characteristic polynomial: p(A) = 0.
The product of all eigenvalues equals the determinant (λ₁·λ₂ = det(A)) and their sum equals the trace (λ₁ + λ₂ = tr(A)). These are Vieta's formulas applied to the characteristic polynomial.
Principal Component Analysis finds directions of maximum variance. It eigendecomposes the covariance matrix: eigenvectors are principal components, eigenvalues quantify variance explained. Sorting by descending eigenvalue ranks the most informative directions.
A defective matrix has fewer linearly independent eigenvectors than its size. For 2×2, this occurs with a repeated eigenvalue having only one eigenvector. Example: [[2,1],[0,2]] — eigenvalue 2 (double) but only eigenvector [1,0]. Defective matrices cannot be diagonalized.
For discrete systems xₙ₊₁ = Axₙ: stable if all |λᵢ| < 1, unstable if any |λᵢ| > 1. For continuous systems ẋ = Ax: stable if all Re(λᵢ) < 0. This underpins control theory and differential equations.
Eigenvalues and eigenvectors are fundamental to linear algebra: given a square matrix A, an eigenvector v is a non-zero vector that satisfies Av = λv, where the scalar λ is its eigenvalue. This means the transformation only stretches or compresses v — it never rotates it away from its original line. Eigendecomposition underpins dimensionality reduction (PCA), Google's PageRank algorithm, quantum mechanical observables, and structural vibration analysis in engineering.
This simulation lets you edit every entry of a 2×2 matrix in real time and watch the unit circle deform into an ellipse as the transformation is applied. Red and blue arrows mark the two eigenvectors, and the HUD displays the trace, determinant, and both eigenvalues (real or complex) as you drag the sliders.
What is an eigenvalue and why does it matter?
An eigenvalue λ of a matrix A is a scalar for which there exists a non-zero vector v with Av = λv. It measures how much the transformation stretches (|λ| > 1), compresses (|λ| < 1), or reflects (λ < 0) space along a specific direction. Eigenvalues appear in quantum mechanics, control theory, image compression, and network analysis — essentially wherever repeated linear maps arise.
How do you compute eigenvalues of a 2×2 matrix?
Solve the characteristic equation det(A − λI) = 0, which for a 2×2 matrix expands to λ² − tr(A)·λ + det(A) = 0. The quadratic formula gives λ = [tr(A) ± √(tr(A)² − 4·det(A))] / 2. If the discriminant tr²− 4·det is negative, eigenvalues are complex conjugates and the transformation contains a rotational component.
What happens to the unit circle under a 2×2 matrix?
A 2×2 matrix always maps the unit circle to an ellipse. For symmetric matrices, the semi-axes of the ellipse point along the eigenvectors and have lengths equal to the absolute eigenvalues. For non-symmetric matrices the axes align with singular vectors instead. Watching the circle deform is a geometric way to see the matrix's action on all directions simultaneously.
When tr(A)² − 4·det(A) < 0, eigenvalues are complex conjugates α ± βi. No real eigenvectors exist; instead the transformation combines rotation with scaling. The modulus |λ| = √(α² + β²) tells you whether points spiral inward (|λ| < 1), outward (|λ| > 1), or trace ellipses (|λ| = 1). Pure rotation matrices have eigenvalues on the unit circle with |λ| = 1.
For any square matrix, the sum of all eigenvalues equals the trace and their product equals the determinant. For a 2×2 matrix: λ₁ + λ₂ = tr(A) and λ₁·λ₂ = det(A). These are Vieta's formulas applied to the characteristic polynomial and give quick sanity checks — if det(A) = 0 then at least one eigenvalue is zero, meaning the transformation collapses a dimension.
A matrix is defective if it has fewer linearly independent eigenvectors than its size. For a 2×2 matrix this occurs when the characteristic polynomial has a repeated root but only one eigenvector direction. The classic example is [[2,1],[0,2]]: eigenvalue 2 (double), but only the eigenvector [1,0]. Defective matrices cannot be diagonalised; they require a Jordan normal form instead.
PCA finds orthogonal directions of maximum variance in a data set. It eigendecomposes the covariance matrix C = (1/n)·XᵀX: the eigenvectors are the principal components (directions of variance), and the eigenvalues quantify exactly how much variance each component captures. Sorting by descending eigenvalue lets you retain only the top k components and still explain most of the data — this is the basis of face recognition, data visualisation, and noise reduction.
For a discrete system xₙ₊₁ = A·xₙ, stability requires all eigenvalue magnitudes |λᵢ| < 1. If any |λᵢ| > 1, the system grows without bound (unstable). For continuous systems ẋ = Ax, the criterion shifts to Re(λᵢ) < 0 for all eigenvalues. These facts underpin the Lyapunov stability theory used in robotics, aircraft autopilots, and economic models.
The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial. If p(λ) = det(A − λI) then p(A) = 0. For a 2×2 matrix this means A² − tr(A)·A + det(A)·I = 0. The theorem is used in control theory to express high powers of A in terms of lower powers, simplifying the analysis of state-space models.
The spectral radius ρ(A) = max|λᵢ| is the largest absolute eigenvalue. It controls the long-term growth rate of iterative maps — repeated multiplication by A amplifies vectors by approximately ρ(A)ⁿ after n steps. In numerical methods, ρ(A) < 1 guarantees convergence of iterative solvers like Jacobi or Gauss-Seidel. The Google PageRank power iteration converges because the link matrix has ρ = 1 with a unique dominant eigenvector.
Yes. If λ = 0 is an eigenvalue, then Av = 0 for some non-zero v, meaning det(A) = 0 and the matrix is singular (not invertible). The null space of A is spanned by the zero-eigenvalue eigenvectors. In geometric terms the transformation collapses at least one dimension — a 2D matrix with a zero eigenvalue squashes the plane onto a line or a point.