Enter Matrix Values
Example Data Table
| Matrix A | λ | Expected idea |
|---|---|---|
5 1 0 0 5 1 0 0 5 |
5 | A Jordan block gives one eigenvector and higher generalized eigenvectors. |
2 1 0 2 |
2 | The second kernel power reveals the full generalized eigenspace. |
3 0 0 4 |
3 | The ordinary eigenspace already contains the needed vector. |
Formula Used
Let A be a square matrix and λ be an eigenvalue. First form B = A − λI. A generalized eigenvector of rank k satisfies:
(A − λI)^k v = 0 and (A − λI)^(k−1) v ≠ 0
The calculator computes the nullspace of B, B², B³, and higher powers. The nullity growth shows how many independent vectors enter each level. New vectors in ker(Bᵏ) but not in ker(Bᵏ⁻¹) are generalized eigenvectors of that level.
Rank and nullity are linked by:
rank(Bᵏ) + nullity(Bᵏ) = n
How to Use This Calculator
- Enter a square matrix. Keep one row per line.
- Enter the eigenvalue λ that you want to test.
- Select a maximum chain order. Use the matrix size for a full check.
- Adjust tolerance only when decimal roundoff affects results.
- Press the submit button.
- Read the result section above the form.
- Use CSV for spreadsheet work.
- Use PDF for saving or printing the result.
Article: Understanding Generalized Eigenvectors
What They Mean
A usual eigenvector points in a direction that keeps its line after a matrix acts on it. It only changes by a scale factor. Generalized eigenvectors extend this idea. They appear when a matrix does not have enough ordinary eigenvectors. This situation is common in Jordan form problems.
Why Kernel Powers Matter
The shifted matrix B = A − λI is the main object. Ordinary eigenvectors live in the kernel of B. Generalized eigenvectors may not live there. Instead, they become zero after B is applied several times. So the calculator checks B, B², B³, and more. Each power can reveal a larger nullspace.
Reading the Growth
The nullity column is important. If nullity increases from one order to the next, new generalized directions appeared. These directions help build Jordan chains. A chain starts with a high order generalized vector. Applying B moves the vector down the chain. Eventually it reaches an ordinary eigenvector. One more application gives the zero vector.
Practical Use
This tool is useful for linear algebra, differential equations, control systems, and matrix decomposition. It shows ranks, nullities, basis vectors, and chain previews. The graph makes the growth easy to see. Flat growth means no new vectors were found at that level. For exact classroom work, use clean integer matrices when possible. For decimal matrices, choose a reasonable tolerance.
Study Tip
Start with small matrices. Compare the first kernel with later kernels. Notice when the generalized eigenspace becomes stable. That stable space is the full generalized eigenspace for the selected eigenvalue.
FAQs
What is a generalized eigenvector?
It is a non-zero vector v where (A − λI)^k v = 0 for some positive integer k. If k is greater than one, it is not just an ordinary eigenvector.
What does the order mean?
The order is the power k used in (A − λI)^k. A larger order checks whether vectors become zero only after repeated matrix action.
Why do I need an eigenvalue?
Generalized eigenvectors are tied to a chosen eigenvalue. The calculator forms A − λI, then studies the kernels of its powers.
What is the first kernel?
The first kernel is ker(A − λI). It contains ordinary eigenvectors. Later kernels may contain additional generalized eigenvectors.
What does nullity growth show?
Nullity growth shows new independent generalized directions. If nullity stops growing, the generalized eigenspace has likely stabilized.
Can this handle decimal matrices?
Yes. Decimal matrices are accepted. Use the tolerance field carefully because very small roundoff errors may affect rank and nullspace detection.
Why is my result empty?
The entered λ may not be an eigenvalue, or the tolerance may be too strict. Check the matrix, eigenvalue, and decimal precision.
What is a Jordan chain?
A Jordan chain links generalized vectors by repeated multiplication with A − λI. It ends at an ordinary eigenvector and then reaches zero.