Enter matrix data
Example data table
| Case | Matrix | Eigenvalue summary | Expected Jordan blocks |
|---|---|---|---|
| Defective 3 × 3 | [5 1 0; 0 5 1; 0 0 5] | λ = 5, algebraic multiplicity 3, geometric multiplicity 1 | [J3(5)] |
| Mixed 4 × 4 | [2 1 0 0; 0 2 0 0; 0 0 3 1; 0 0 0 3] | λ = 2 and 3, each with algebraic multiplicity 2 | [J2(2)] ⊕ [J2(3)] |
| Diagonalizable 3 × 3 | [1 0 0; 0 2 0; 0 0 3] | Three distinct eigenvalues | diag(1, 2, 3) |
Formula used
Characteristic polynomial: p(λ) = det(λI - A)
Algebraic multiplicity: the power of (λ - λi) in p(λ)
Geometric multiplicity: dim Ker(A - λI)
Nullity growth: nk = dim Ker(A - λI)k
Blocks of size at least k: dk = nk - nk-1
Blocks of exact size k: bk = dk - dk+1
The calculator builds each Jordan block Js(λ) with λ on the diagonal and ones on the superdiagonal. The full Jordan matrix is the direct sum of all detected blocks.
How to use this calculator
- Choose the matrix size from 2 × 2 up to 4 × 4.
- Paste the matrix entries with one row per line.
- Set a tolerance when eigenvalues are repeated or nearly repeated.
- Choose the display precision for matrix and eigenvalue output.
- Press Calculate Jordan Form to display the result above the form.
- Review eigenvalue multiplicities, nullity growth, and Jordan block sizes.
- Use CSV or PDF export to save the report.
FAQs
1) What does the Jordan form show?
It shows how a matrix decomposes into Jordan blocks. Each block reveals an eigenvalue and whether the matrix has missing eigenvectors or a complete eigenbasis.
2) When is a matrix diagonalizable?
A matrix is diagonalizable when every Jordan block has size one. Equivalently, geometric multiplicity equals algebraic multiplicity for each eigenvalue.
3) Why are nullities of powers important?
They reveal how many Jordan blocks have length at least one, two, three, and so on. That growth determines the exact block-size pattern.
4) Why can repeated eigenvalues be difficult numerically?
Repeated roots are sensitive to rounding. Small perturbations can split a repeated eigenvalue into nearby estimates, which is why tolerance settings matter.
5) Does this calculator support complex eigenvalues?
Yes. The calculator approximates complex eigenvalues and builds the Jordan form over the complex field when nonreal eigenvalues are detected.
6) Can I use decimal entries?
Yes. Real decimal entries are accepted. Exact integers often produce cleaner Jordan structure because they reduce numerical ambiguity near repeated roots.
7) Why is the result shown above the form?
That layout lets you see the computed Jordan matrix immediately after submission without scrolling past the entire input area again.
8) What matrices can I enter here?
Enter any real square matrix from 2 × 2 to 4 × 4. The tool then estimates eigenvalues, multiplicities, and Jordan blocks numerically.