This implementation computes the Jordan decomposition for 2×2 matrices over ℂ, including distinct, repeated, defective, and complex-eigenvalue cases.
| Example Matrix | Eigenvalues | Jordan Form | Interpretation |
|---|---|---|---|
| [[5, 1], [0, 5]] | 5, 5 | [[5, 1], [0, 5]] | Defective matrix with one Jordan block of size 2. |
| [[4, 1], [2, 3]] | 5, 2 | diag(5, 2) | Two distinct real eigenvalues, so the matrix is diagonalizable. |
| [[0, -1], [1, 0]] | i, -i | diag(i, -i) | Diagonalizable over ℂ with complex conjugate eigenvalues. |
1. For a 2×2 matrix A = [[a, b], [c, d]], compute:
Trace: tr(A) = a + d
Determinant: det(A) = ad - bc
Characteristic polynomial: p(λ) = λ² - tr(A)λ + det(A)
Discriminant: Δ = tr(A)² - 4det(A)
2. If Δ > 0, then A has two distinct real eigenvalues and J = diag(λ1, λ2).
3. If Δ = 0 and A = λI, then J = diag(λ, λ).
4. If Δ = 0 and A ≠ λI, then J = [[λ, 1], [0, λ]].
5. If Δ < 0, the Jordan form is diagonal over ℂ with conjugate eigenvalues.
6. The decomposition is reconstructed by A = PJP-1.
- Enter the four entries of your 2×2 matrix.
- Set the decimal precision and numerical tolerance.
- Choose a preset if you want to test a known matrix type.
- Press the calculate button.
- Review the eigenvalues, Jordan form, basis matrix, inverse, and reconstruction check.
- Inspect the Plotly graph to see eigenvalue locations on the complex plane.
- Download the summary as CSV or PDF when needed.
1. What does Jordan decomposition show?
It rewrites a matrix as A = PJP-1, where J contains Jordan blocks. This reveals eigenvalues, algebraic structure, and whether the matrix is diagonalizable or defective.
2. What is a Jordan block?
A Jordan block places one eigenvalue on the diagonal and ones on the superdiagonal. It appears when a repeated eigenvalue does not have enough independent eigenvectors.
3. When is the matrix diagonalizable?
A 2×2 matrix is diagonalizable when it has two independent eigenvectors. That happens with distinct eigenvalues, or with a repeated eigenvalue when the matrix is already λI.
4. What does defective mean here?
A defective matrix has fewer eigenvectors than its algebraic multiplicity suggests. In 2×2 cases, that means one repeated eigenvalue but only one independent eigenvector.
5. Why does the calculator use a tolerance?
Floating-point arithmetic can create tiny rounding noise. Tolerance helps classify near-zero values correctly when checking repeated eigenvalues, singularity, and reconstruction quality.
6. Can this calculator handle complex eigenvalues?
Yes. When the discriminant is negative, the calculator builds the Jordan form over ℂ and plots the conjugate pair on the complex plane.
7. What does reconstruction error mean?
It measures the gap between the original matrix and PJP-1. A very small value indicates the computed decomposition reconstructs the matrix accurately.
8. Does this calculator support larger matrices?
This implementation focuses on 2×2 matrices for reliability and clarity. The same principles extend to larger matrices, but the computations become more involved.