Eigenvectors Calculator

Compute eigenvalues and eigenvectors for 2x2 to 5x5 matrices. Handles real and complex numbers, with 2D visualization for real eigenpairs.

Definition: For a square matrix A, an eigenvector v satisfies A·v = λ·v, where λ is the corresponding eigenvalue.

2x2 : [[2,1],[1,2]]
[[1,2],[2,1]]
Rotation 90° (complex)
3x3 diagonal
Symmetric 3x3

Understanding Eigenvalues and Eigenvectors

In linear algebra, eigenvectors are non-zero vectors that change only by a scalar factor when a linear transformation is applied. The corresponding eigenvalue indicates the factor by which the eigenvector is scaled.

Mathematical definition: A·v = λ·v, where A is a square matrix, v is the eigenvector, and λ is the eigenvalue.

Characteristic equation: det(A - λI) = 0, which yields the eigenvalues.

Properties

  • Eigenvectors corresponding to distinct eigenvalues are linearly independent.
  • A matrix is diagonalizable if it has a full set of linearly independent eigenvectors.
  • Symmetric matrices always have real eigenvalues and orthogonal eigenvectors.
  • The trace equals the sum of eigenvalues, the determinant equals the product.

How the Calculator Works

1
Select matrix dimension (2x2 or 3x3) and enter the entries.
2
Click "Compute Eigen". The calculator uses math.js to compute eigenvalues and eigenvectors numerically.
3
Results are displayed: eigenvalues (may be complex) and corresponding eigenvectors.
4
For 2x2 matrices with real eigenvalues, the eigenvectors are visualized as arrows from the origin.

Applications

  • Physics: Principal axes of rotation, quantum mechanics.
  • Engineering: Vibration analysis, stability of systems.
  • Data Science: Principal Component Analysis (PCA).
  • Computer Graphics: Understanding transformations.

Calculator Features:

  • Supports 2x2 and 3x3 matrices with real or complex entries.
  • Handles complex eigenvalues and eigenvectors (displayed in a+bi format).
  • Visualizes real eigenvectors for 2D (when eigenvalues are real).
  • Pre-loaded examples for quick testing.

Frequently Asked Questions

Complex eigenvalues indicate rotational components in the transformation. The eigenvectors are also complex and cannot be visualized in real 2D/3D space. They often occur in pairs (conjugates).

Yes, you can enter integers, decimals, or even expressions like 1/2 (the calculator uses JavaScript evaluation, so 1/2 will be interpreted as 0.5). For fractions, use decimal notation or compute manually.

The calculator uses math.js which performs numerical computations with double precision. For most purposes it's accurate, but be aware of floating-point errors for ill-conditioned matrices.

Any scalar multiple of an eigenvector is also an eigenvector. The calculator returns normalized vectors (not necessarily unit length, but scaled to have largest component 1, and may show signs). Different algorithms may give different multiples.