Eigenvalue Calculator

Compute eigenvalues, eigenvectors, characteristic polynomial, trace, determinant. Visualize real eigenvectors for 2x2 matrices; full analytical results for 3x3. Perfect for PCA, quantum mechanics, and engineering stability.

Enter 2x2 Matrix A
Matrix A = [[a₁₁, a₁₂], [a₂₁, a₂₂]]
? Symmetric (2,1;1,2)
? Diagonal (3,0;0,5)
? Rotation 90° (0,-1;1,0)
✂️ Shear (1,2;0,1)
⚠️ Singular (2,4;1,2)
⭐ Identity (1,0;0,1)
? 3x3 Diagonal (3,0,0;0,5,0;0,0,2)
? 3x3 Symmetric (2,1,0;1,2,0;0,0,3)
? 3D Rotation (1,0,0;0,0,-1;0,1,0)
Zero data persistence – All matrix calculations run locally inside your browser. No information is uploaded.

What are Eigenvalues and Eigenvectors?

In linear algebra, an eigenvector of a square matrix A is a non-zero vector v such that A v = λ v, where λ (lambda) is a scalar known as the eigenvalue. Geometrically, eigenvectors represent directions that remain invariant under the linear transformation — they may be stretched, compressed, or reversed, but not rotated out of their span.

A · v = λ v ⇔ (A - λI) v = 0

The characteristic equation det(A - λI) = 0 yields eigenvalues. For a 2×2 matrix, the characteristic polynomial is λ² − (tr A)λ + det(A) = 0. For 3×3, it is -λ³ + tr(A) λ² - (sum of principal minors) λ + det(A)=0.

Eigenvectors are fundamental to quantum mechanics (states & observables), Google's PageRank algorithm, facial recognition (Eigenfaces), vibration analysis, and principal component analysis (PCA) in machine learning.

Step-by-step algorithm used by this calculator

  • 2x2: Characteristic λ² − τλ + Δ = 0 → eigenvalues via quadratic formula. Eigenvectors via direct formula.
  • 3x3: Compute coefficients: a = -1, b = tr(A), c = -(M11+M22+M33), d = det(A). Solve cubic using Cardano’s method with numerical safeguards. For each real eigenvalue, solve (A-λI)v = 0 via Gaussian elimination with partial pivoting.
  • Normalization: Unit vectors are computed for clean comparison.
  • Verification: Residual norm ||Av - λv|| is displayed for each eigenpair (typical < 1e-12).
Verified accuracy: After each calculation, the tool displays the residual norm ||A·v − λ·v|| for each eigenpair. Typical residuals are below 10−12, confirming the precision of our solver.
Real‑world case: PCA & Dimensionality reduction

In data science, eigenvectors of the covariance matrix reveal principal components. The eigenvector associated with the largest eigenvalue indicates the direction of maximum variance. This calculator helps users comprehend how 2D point clouds stretch, which directly translates to understanding feature importance in machine learning pipelines. For 3x3, the same principle applies to three-dimensional data.

Geometric Interpretation & Classification

Matrix type Eigenvalue nature Transformation behavior
Symmetric Real, orthogonal eigenvectors Stretch along orthogonal axes
Rotation (2D) Complex conjugate Pure rotation, no real eigenvectors
Defective (repeated eigenvalue) May have fewer independent eigenvectors Shear-like behavior

Common Misconceptions & Expert Clarifications

  • Only square matrices have eigenvectors – Correct; our tool focuses on 2×2 and 3×3.
  • Eigenvectors are always orthogonal – Only for symmetric (or Hermitian) matrices.
  • Every matrix has real eigenvectors – No: rotation matrices produce complex eigenvectors.
  • Numerical stability – For matrices near singularity, small rounding errors may appear. Our algorithm includes explicit tolerance checks.

Academic reliability & peer-inspired standards – This implementation follows canonical linear algebra theory (Strang, 2019; Meyer, 2000). The eigen-solving method is numerically stable for 2x2 and 3x3 matrices and has been cross‑validated against symbolic solvers (SymPy, Wolfram Alpha).
Review process: The calculator is maintained by the GetZenQuery tech team, with regular audits every software release. Last verification: May 2026. 

Frequently Asked Questions

Complex eigenvalues indicate that the linear transformation includes a rotation component. For 2x2 no real eigenvectors; for 3x3 a real eigenvector may still exist for the real eigenvalue.

Double-precision, residuals < 1e-12. Verification panel shows actual ||Av - λv||.

Currently only 2x2 and 3x3. Higher dimensions require more advanced numerical methods; future updates may include larger sizes.
References: MathWorld: Eigenvalue, Strang, G. "Introduction to Linear Algebra", 5th ed., Wikipedia: Eigenvalues.