Enter Matrix and Vector Values
Example Data Table
| Example | Matrix A | Eigenvalues | Eigenvectors | Input Vector | Transformed Vector |
|---|---|---|---|---|---|
| Diagonal stretch | [[3, 0], [0, 2]] | 3, 2 | [1, 0], [0, 1] | [1, 1] | [3, 2] |
| Distinct directions | [[4, 1], [2, 3]] | 5, 2 | [0.707107, 0.707107], [0.447214, -0.894427] | [1, 0] | [4, 2] |
| Complex pair | [[0, -1], [1, 0]] | i, -i | No real eigenvectors | [1, 0] | [0, 1] |
Formula Used
Characteristic polynomial: λ2 − (trace A)λ + det(A) = 0
Trace: trace(A) = a + d
Determinant: det(A) = ad − bc
Discriminant: Δ = (trace A)2 − 4 det(A)
Eigenvalues: λ = ((trace A) ± √Δ) / 2
Eigenvectors: Solve (A − λI)v = 0 for each eigenvalue.
Vector transformation: Av = [ax + by, cx + dy]
Geometric meaning: Eigenvectors keep their direction line, while eigenvalues show scaling or reflection along that line.
How to Use This Calculator
- Enter the four values of your 2×2 matrix.
- Provide a test vector to see how the matrix moves it.
- Set the graph extent for a wider or tighter view.
- Choose plot samples for smoother transformed curves.
- Press Visualize Eigenvectors to generate results.
- Read the eigenvalues, eigenvectors, and classification cards.
- Inspect the graph to compare original and transformed shapes.
- Export the summary with the CSV or PDF buttons.
Frequently Asked Questions
1. What does this calculator visualize?
It visualizes how a 2×2 matrix transforms vectors, the unit circle, and the unit square. It also highlights real eigenvector lines whenever they exist.
2. Why are some eigenvectors missing?
If the matrix has complex eigenvalues, there are no real eigenvectors in the plane. The graph still shows the transformation, but no real eigendirection lines appear.
3. What is the meaning of an eigenvalue?
An eigenvalue tells how much the matrix stretches, shrinks, or flips an eigenvector. Positive values preserve direction, while negative values reverse it.
4. Why is the unit circle useful here?
The unit circle shows how the matrix acts on many directions at once. Its image reveals rotation, shear, anisotropic scaling, and distortion patterns clearly.
5. What happens when the determinant is zero?
A zero determinant means the matrix collapses area and becomes singular. At least one direction is flattened, so the transformation cannot be inverted.
6. Why can a repeated eigenvalue have one eigenvector?
Some repeated eigenvalues produce only one independent eigendirection. That happens for defective matrices, where the algebraic multiplicity exceeds the geometric multiplicity.
7. Are the displayed eigenvectors normalized?
Yes. The calculator normalizes displayed eigenvectors to unit length. That makes comparison easier, while preserving the correct eigendirection on the graph.
8. Can I use this for larger matrices?
This page is designed for 2×2 matrices because they can be shown directly in a plane. Larger matrices need different visual methods.