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Premium experiment • invariant directions

MATHEMATICS • LINEAR ALGEBRA

Eigenvalues & Eigenvectors

Edit a real 2×2 matrix and a test vector to see invariant directions, scaling factors, trace, determinant and the characteristic polynomial on the same plane.

● MATRIX PLANEEigenvalues & Eigenvectors
Eigenvalue λ₁
Eigenvalue λ₂
Determinant
Trace

Reasoning inspector

live model state
Trace—
Determinant—
Characteristic discriminant—
λ₁—
λ₂—
|Av| / |v|—
v–Av line alignment—
Eigenvalue type—

See invariant directions

For a real eigenvector v, multiplying by the matrix keeps the result on the same line: Av = λv. Matrices with a negative discriminant instead produce a complex-conjugate eigenvalue pair.

LEARN WITH THE SIMULATION

Understand Eigenvalues & Eigenvectors by experimenting.

START HERE

Eigenvectors are invariant directions of a linear transformation: applying A to an eigenvector v keeps the result Av on the same line, changing only its scale and possibly reversing direction. This laboratory connects a real 2×2 matrix to trace, determinant, characteristic discriminant, eigenvalues, displayed eigendirections and a test vector. The premium challenge measures the alignment angle between v and Av so the defining equation Av=λv becomes a geometric experiment.

BUILD THE IDEA

What you should understand first

  • Compute eigenvalues from λ²−tr(A)λ+det(A)=0 for a 2×2 matrix.
  • Interpret a real eigenvector as a direction whose image remains collinear with the original vector.
  • Relate |Av|/|v| to eigenvalue magnitude when v is on an eigendirection.
  • Recognize matrices with complex-conjugate eigenvalues and therefore no real eigendirections.

TRY IT YOURSELF

How can you recognize an eigenvector geometrically without first solving all of the algebra?

  1. Choose a matrix with two real eigendirections and place the test vector away from them.
  2. Move the vector toward one displayed eigendirection while watching v and Av.
  3. Reach the alignment tolerance and compare |Av|/|v| with the corresponding eigenvalue magnitude.
  4. Change the matrix until the characteristic discriminant becomes negative and observe the disappearance of real eigendirections.
Read the visual:

For a real eigenvector, v and Av are parallel or antiparallel. Their length ratio gives |λ|, while a negative eigenvalue reverses direction. A negative characteristic discriminant produces a complex-conjugate pair instead of real eigenvector lines.

WHY IT WORKS 01

Trace and determinant encode the two eigenvalues

For a 2×2 matrix, the characteristic polynomial is λ²−tr(A)λ+det(A). The eigenvalue sum equals the trace and their product equals the determinant. This lets the inspector connect geometric behavior to two compact matrix invariants rather than treating eigenvalues as disconnected calculator outputs.

WHY IT WORKS 02

An eigendirection converts a matrix action into scalar multiplication

Most vectors are both rotated and rescaled by a matrix. Along an eigenvector, the transformation simplifies to Av=λv. The direction remains on one line, and the scale factor is λ; negative λ places Av on the opposite ray. This is why repeated applications Aⁿ become especially simple on eigenvectors.

WHY IT WORKS 03

Complex eigenvalues often signal rotation-like planar behavior

When the characteristic discriminant is negative, a real 2×2 matrix has a complex-conjugate eigenvalue pair and no nonzero real vector satisfying Av=λv with real λ. Rotations through nonzero non-180° angles are a familiar example: every real direction is turned away from itself.

DO NOT MIX THESE UP

Common misconception

Eigenvectors are not generally matrix columns, and a repeated eigenvalue does not guarantee two independent eigenvectors. Parallel and antiparallel transformed vectors both satisfy collinearity, with the sign of λ distinguishing whether direction is preserved or reversed. Complex eigenvalues of a real 2×2 matrix do not imply the matrix entries themselves are complex.

PRACTICE CHALLENGE

Can you align the test vector with an eigendirection and verify the eigenvalue using only the resulting geometry?

  • Choose an alignment tolerance.
  • Move the test vector until v and Av are collinear within that tolerance.
  • Compare the inspector's vector-scale ratio with one eigenvalue magnitude.
  • Determine from the drawing whether λ is positive or negative by checking whether Av points along or opposite v.
Reveal the reasoning

When v is an eigenvector, Av=λv. Therefore v and Av lie on the same line, |Av|/|v|=|λ|, and their orientation reveals the sign: same ray means λ>0, opposite rays mean λ<0. This test applies only to real eigendirections.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • The transformation is a finite real 2×2 matrix acting on vectors in the ordinary Euclidean plane.
  • Eigenvalues are computed from the quadratic characteristic polynomial using browser floating-point arithmetic.
  • Displayed real eigenvectors are normalized directions and either sign represents the same invariant line.

Model limitations

  • The simulator does not compute generalized eigenvectors, Jordan chains or eigenspaces for matrices larger than 2×2.
  • Matrices with complex eigenvalues are reported as a conjugate pair without drawing complex eigenvectors in a higher-dimensional representation.

CONNECT IT

Where this matters

  • Dynamical-system stability
  • Modal and vibration analysis
  • Principal transformation directions
  • Differential equations and repeated matrix powers

WORDS THAT MATTER

Key vocabulary

Eigenvector
Nonzero vector v whose image satisfies Av=λv for some scalar λ.
Eigenvalue
Scalar λ giving the signed scaling of its associated eigenvector.
Trace
Sum of diagonal matrix entries and also sum of eigenvalues counting multiplicity.
Determinant
Signed area-scale factor of a 2D linear map and product of its eigenvalues.
Characteristic discriminant
tr(A)²−4det(A), whose sign determines whether a real 2×2 matrix has real or complex-conjugate eigenvalues.