🧮 Eigenvectors & Eigenvalues
Adjust any 2×2 matrix and see its eigenvectors — the special directions that a linear transformation only stretches, never rotates. Watch the characteristic polynomial solve for λ, the unit circle deform into an ellipse, and eigenvectors align with its principal axes.
📐 Matrix A
🎯 Presets
λ Eigenvalues
📊 Properties
det(A − λI) = 0
λ² − tr(A) λ + det(A) = 0
Av = λv (v ≠ 0)
Eigenvectors span the eigenspace.
For symmetric A: eigenvectors are orthogonal.
Key ideas
Eigenvectors are the vectors
v that satisfy
A v = λ v for a scalar λ (the
eigenvalue). Geometrically, they are the directions the
transformation only stretches or flips — never
rotates. Every point on the green/purple dashed lines is an
eigenvector.
The characteristic polynomial λ²
− tr(A)λ + det(A) = 0 gives the eigenvalues. Its
discriminant Δ = tr² − 4 det determines the type:
Δ > 0 → two distinct real eigenvalues; Δ = 0 →
repeated; Δ < 0 → complex conjugates (pure rotation has no
real eigenvectors).
The yellow ellipse is the image of the unit circle
under A. Its semi-axes are exactly the eigenvectors scaled by
|λ|.
Related: Matrix Transformations →
🧮 Eigenvectors & Eigenvalues
About this simulation
Every 2×2 matrix represents a linear transformation of the plane, and its eigenvectors are the rare directions that the transformation only stretches or flips — never rotates. This visualizer lets you adjust the four matrix entries and instantly see the eigenvalues, eigenvector lines, and how the unit circle morphs into an ellipse. Eigenanalysis powers Google's PageRank, principal component analysis (PCA), vibration modes in engineering, and quantum mechanics, making it one of the most useful ideas in all of mathematics.
How it works
- Move the four sliders a, b, c, d to define matrix A.
- The characteristic polynomial λ² − tr(A)λ + det(A) = 0 is solved for the eigenvalues.
- Each eigenvalue's null space (A − λI)v = 0 gives an eigenvector direction, drawn as a dashed line.
- The yellow ellipse is the image of the unit circle; its semi-axes line up with the eigenvectors scaled by |λ|.
Key equations
A v = λ v — v is the eigenvector, λ
its eigenvalue. They are found from
det(A − λI) = 0, which expands to
λ² − tr(A)λ + det(A) = 0 where
tr(A) = a+d and det(A) = ad−bc.
Controls
- a, b, c, d sliders: set each matrix entry.
- Presets: jump to identity, scaling, rotation, shear, symmetric, reflection and more.
- Animate transform: morph smoothly between the original and transformed grid.
- Reset: snap back to the fully transformed view.
Did you know?
When the discriminant tr² − 4·det is negative the eigenvalues become complex conjugates — the matrix rotates the plane, so there is no real direction left unturned. A pure rotation has no real eigenvectors at all.
About Eigenvectors & Eigenvalues Visualizer
This simulation lets you explore eigenvectors and eigenvalues of any 2×2 matrix by adjusting four sliders in real time. An eigenvector of a matrix A is a non-zero vector v that satisfies Av = λv, meaning the transformation only scales v by the scalar eigenvalue λ without changing its direction. The visualizer solves the characteristic polynomial, draws the eigenvector lines on a coordinate grid, and shows how the unit circle deforms into an ellipse whose principal axes align with the eigenvectors.
Eigenanalysis is one of the most broadly applied ideas in mathematics and science, underpinning Google PageRank, principal component analysis (PCA) in data science, vibration mode analysis in structural engineering, and the Schrodinger equation in quantum mechanics.
Frequently Asked Questions
What is an eigenvector?
An eigenvector of a square matrix A is a non-zero vector v such that Av = λv for some scalar λ called the eigenvalue. Geometrically, applying the matrix transformation to an eigenvector only stretches or flips it along its own direction; it is never rotated to point somewhere else. Every scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue, so eigenvectors define directions, not specific vectors.
How do I use this simulation?
Drag the four sliders labelled a, b, c, d to set the entries of the 2×2 matrix. The canvas instantly redraws the eigenvector lines (green and purple dashed), the transformed basis vectors, and the yellow ellipse that is the image of the unit circle. Use the preset buttons to jump to classic examples such as a pure rotation (which has no real eigenvectors) or a symmetric matrix (whose eigenvectors are guaranteed to be perpendicular). Click "Animate transform" to watch the transformation morph smoothly from the identity to the full matrix A.
When does the matrix have no real eigenvectors?
A 2×2 matrix has real eigenvectors only when the discriminant of the characteristic polynomial, tr(A)^2 minus 4 det(A), is greater than or equal to zero. When this discriminant is negative the two eigenvalues form a complex conjugate pair and the transformation is a rotation with scaling; no real direction stays fixed. You can observe this directly by moving the sliders toward the "Rotate 90 deg" preset, at which point the simulation displays the "Complex eigenvalues" note and the dashed eigenvector lines disappear.
How are eigenvalues computed from the matrix entries?
For a 2×2 matrix with entries a, b, c, d, you first form the characteristic equation det(A minus lambdaI) = 0. Expanding the determinant gives the quadratic lambda^2 minus tr(A)*lambda + det(A) = 0, where tr(A) = a+d is the trace and det(A) = ad minus bc is the determinant. Applying the quadratic formula yields lambda = (tr(A) plus or minus sqrt(tr(A)^2 minus 4*det(A))) / 2. Once each eigenvalue is known, the corresponding eigenvector is found from the null space of (A minus lambdaI), i.e. by solving (A minus lambdaI)v = 0 for a non-zero v.
How are eigenvectors used in principal component analysis (PCA)?
PCA reduces the dimensionality of a dataset by finding the directions along which the data varies most. Those directions are precisely the eigenvectors of the data covariance matrix, and the corresponding eigenvalues measure the amount of variance captured along each direction. By keeping only the eigenvectors with the largest eigenvalues you project the data onto a lower-dimensional subspace that retains the most information. The "PCA-like" preset in this simulation shows a covariance-style symmetric matrix where the two eigenvectors are perpendicular and the eigenvalues indicate relative spread.
Is it true that eigenvectors of a symmetric matrix are always perpendicular?
Yes. This is guaranteed by the Spectral Theorem for real symmetric matrices: if A equals A transposed, then all eigenvalues are real and eigenvectors belonging to distinct eigenvalues are orthogonal to each other. You can verify this in the simulation by selecting the "Symmetric" or "PCA-like" presets and observing that the two dashed eigenvector lines meet at a right angle. This property is what makes symmetric matrices so powerful in physics and data analysis, where covariance matrices and Hamiltonians are both symmetric (or Hermitian in the complex case).
Who discovered eigenvectors and eigenvalues?
The concept grew out of 18th-century work on the rotation of rigid bodies and the solution of differential equations. Euler studied principal axes of rotation in 1758, and Lagrange encountered characteristic equations while analyzing quadratic forms in mechanics around 1773. The word "eigenvalue" comes from the German word "eigen" (meaning own or characteristic), introduced in the early 20th century by David Hilbert in the context of integral equations. The modern algebraic formulation in terms of the characteristic polynomial and linear operators was consolidated by Augustin-Louis Cauchy and later by Weierstrass and Kronecker in the 19th century.
What other simulations are related to eigenvectors?
The Matrix Transformations simulator on this site is the closest relative, showing how general 2D linear maps distort the plane. Singular value decomposition (SVD) extends eigenvector ideas to non-square matrices and is used in image compression and recommendation systems. Markov chain simulations rely on the dominant eigenvector (the stationary distribution) of a transition matrix. In physics, the normal modes of a vibrating system are the eigenvectors of the system's stiffness matrix, which you can explore in spring-mass or wave simulations.
How are eigenvalues used in structural engineering and vibration analysis?
When engineers model a bridge, aircraft wing, or building frame they assemble a stiffness matrix K and a mass matrix M. The natural vibration frequencies of the structure are the square roots of the eigenvalues of the generalized problem Kv = lambda*Mv, and the corresponding eigenvectors describe the mode shapes in which the structure oscillates. Designing so that operating frequencies stay far from these natural frequencies prevents resonance failures, as famously illustrated by the Tacoma Narrows Bridge collapse in 1940. Modern finite-element packages solve millions of such eigenvalue problems for large structures.
What are open research questions involving eigenvalues?
For large sparse matrices arising in machine learning, quantum chemistry, and network analysis, computing even a handful of the largest or smallest eigenvalues efficiently is an active area of research. Randomized numerical linear algebra, including the randomized SVD and Nystrom approximation, achieves near-optimal eigenvalue estimates in sublinear time on massive matrices. In quantum computing, the quantum phase estimation algorithm offers exponential speedups for certain eigenvalue problems. Another frontier is non-Hermitian physics, where matrices representing open quantum systems have complex eigenvalues whose geometry in the complex plane controls exotic phenomena such as exceptional points and non-Hermitian skin effects.