Eigenvector Simulator

Visualize eigenvalues and transformed vector directions for a 2x2 matrix

Eigenvalue Formulas

Trace	tr(A) = a + d
Determinant	det(A) = ad - bc
Characteristic equation	lambda^2 - tr(A)lambda + det(A) = 0

How to Read the Simulator

Blue directions show approximate eigenvector lines when eigenvalues are real.
Red transformed vectors show how sample directions move under the matrix.
If eigenvalues are complex, the transformation behaves more like a rotation.
Trace and determinant control the characteristic equation.

Example: matrix [[2,1],[1,2]] has eigenvalues 3 and 1. The eigenvector directions are along y = x and y = -x.

Frequently Asked Questions

What is an eigenvector?▼

An eigenvector is a nonzero vector whose direction stays the same after a linear transformation, though its length may scale.

What is an eigenvalue?▼

An eigenvalue is the scale factor applied to an eigenvector by the matrix transformation.

How are eigenvalues found for a 2x2 matrix?▼

They are roots of the characteristic equation lambda^2 - trace(A)lambda + det(A) = 0.

What if eigenvalues are complex?▼

Complex eigenvalues usually indicate rotation-like behavior in a real 2D transformation.

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