Eigenvalue & Eigenvector Visualizer

Compute eigenvalues and eigenvectors of 2x2 and 3x3 matrices. Visualize eigenvectors as invariant directions under transformation.

  • Math & Calculators
  • Runs in your browser
  • Free, no signup

The guide

Tested tool guide Tested browser tools Checked August 16, 2026

What Eigenvalue & Eigenvector Visualizer does, with a checked example

Enter a 2x2 or 3x3 matrix to obtain its eigenvalues and corresponding eigenvectors, then view the real eigenvector directions that remain invariant under the matrix transformation. The results connect the equation Av = lambda v with the geometry of stretching, shrinking, or reversing vectors. A common surprise is that an eigenvector is not unique: every nonzero scalar multiple represents the same direction, so a displayed vector may look different from an equally correct hand-calculated answer.

Worked example

A concrete input and expected output from the current implementation.

Input

2 0
0 3

Expected output

Eigenvalues: {2, 3}. Eigenvector directions: span((1, 0)^T) for eigenvalue 2 and span((0, 1)^T) for eigenvalue 3.

Multiplying (1, 0)^T by the matrix gives 2(1, 0)^T, while multiplying (0, 1)^T gives 3(0, 1)^T. The x-axis and y-axis are therefore invariant directions, scaled by factors of 2 and 3 respectively.

How the result is produced

1

Eigenpair calculation

An eigenpair must satisfy Av = lambda v for a nonzero vector v. The eigenvalues can be checked as the roots of det(A - lambda I) = 0. For each eigenvalue, its eigenvectors form the nonzero part of the null space of A - lambda I. A repeated eigenvalue may have one or several linearly independent eigenvectors.

2

Invariant-direction view

For a real eigenpair, the line through v remains the same line after transformation because Av is a scalar multiple of v. A positive eigenvalue preserves the vector's orientation, a negative eigenvalue reverses it, and zero collapses it to the origin. The magnitude of the eigenvalue gives the scale factor along that direction.

Good uses

  • Checking the eigenvalues and eigenvectors from a hand-solved 2x2 characteristic polynomial exercise.
  • Seeing why a shear matrix has an invariant direction even though most vectors change direction.
  • Comparing the invariant axes and scale factors of candidate 3x3 linear transformations.

Limits and checks

  • Eigenvectors may be normalized, negated, or scaled differently from another source. Compare their directions or verify Av = lambda v instead of comparing coordinates literally.
  • A repeated eigenvalue does not guarantee several independent eigenvectors. One displayed direction may represent only part of the multiplicity information.
  • Complex eigenvectors are not real invariant arrows. If a real matrix has complex eigenpairs, a real-coordinate visualization cannot depict those pairs as ordinary directions.

Common questions

Why does the visualizer show the negative of my eigenvector?

Both vectors are correct. If v satisfies Av = lambda v, then every nonzero multiple cv also satisfies the equation for the same eigenvalue. In particular, v and -v point along the same invariant line but have opposite arrow orientations. Normalized output can also replace your vector with a shorter or longer representative.

Does every 2x2 matrix have a real eigenvector to display?

No. The real 90-degree rotation matrix with rows (0, -1) and (1, 0) has eigenvalues i and -i, so it has no nonzero real eigenvector and no real invariant line. Its eigenpairs exist over the complex numbers, but those complex directions cannot be represented as ordinary arrows in a real two-dimensional plot.

References and verification

The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.