Calculator Input
Example Data Table
| Matrix A | λ | A − λI | Eigenspace Basis | Meaning |
|---|---|---|---|---|
| [[2, 1], [0, 2]] | 2 | [[0, 1], [0, 0]] | {[1, 0]} | One free variable creates one basis vector. |
| [[3, 0], [0, 3]] | 3 | [[0, 0], [0, 0]] | {[1, 0], [0, 1]} | The whole plane is the eigenspace. |
| [[4, 1, 0], [0, 4, 0], [0, 0, 2]] | 4 | [[0, 1, 0], [0, 0, 0], [0, 0, -2]] | {[1, 0, 0]} | The basis has one vector. |
Formula Used
Eigenspace formula:
Eλ(A) = Null(A − λI)
System to solve:
(A − λI)x = 0
Rank-nullity relation:
Nullity(A − λI) = n − Rank(A − λI)
The calculator subtracts λ from each diagonal entry. It row-reduces the new matrix. Pivot columns and free variables are identified. Each free variable produces one basis vector.
How to Use This Calculator
- Select the matrix size.
- Enter every matrix value. Fractions like
3/4are allowed. - Enter the eigenvalue λ.
- Choose tolerance, precision, output style, and vector scaling.
- Press the calculate button.
- Read the rank, nullity, RREF, pivot columns, and basis vectors.
- Use the CSV or PDF buttons to save the result.
Understanding the Basis of an Eigenspace
What the Calculator Finds
A basis of an eigenspace is a compact list of vectors. These vectors describe every eigenvector for one eigenvalue. The calculator starts with a square matrix and a selected eigenvalue. It builds the matrix A minus lambda times the identity matrix. Then it solves a homogeneous linear system. The solutions form the eigenspace.
Why Row Reduction Matters
Row reduction is the main work. It turns the matrix into reduced row echelon form. This form shows pivot columns and free columns. Pivot columns control dependent variables. Free columns create independent directions. Each independent direction becomes one basis vector. This is why nullity is important. A nullity of two means the eigenspace has two basis vectors.
Using the Result Carefully
The determinant check helps confirm the chosen value. If det(A minus lambda I) is not close to zero, the selected value is probably not an eigenvalue. In that case, the only solution may be the zero vector. The zero vector is not used as a basis vector.
Advanced Output Options
Decimal output is useful for measured data. Fraction output is better for classroom work. Integer-like scaling can make vectors easier to read. Unit scaling is helpful when length comparison matters. The residual check compares Av with lambda v. A small residual means the vector passes the eigenvector test. This makes the answer easier to trust.
Practical Learning Value
This tool is useful for linear algebra, matrix theory, engineering models, and data transformations. It does not only give the final basis. It also shows the row-reduction structure. That helps students understand why the answer is correct.
FAQs
1. What is an eigenspace?
An eigenspace is the set of all vectors that satisfy Av = λv for a chosen eigenvalue. It also includes the zero vector.
2. What is a basis of an eigenspace?
It is a smallest independent vector set that spans the whole eigenspace. Every eigenvector can be built from those basis vectors.
3. Why do we solve A − λI?
The equation Av = λv becomes Av − λv = 0. That simplifies to (A − λI)v = 0.
4. What if no basis vector appears?
The selected value may not be an eigenvalue. It can also happen when numerical tolerance is too strict for decimal entries.
5. What does nullity mean here?
Nullity is the number of free variables in A − λI. It equals the number of basis vectors for the eigenspace.
6. Can I enter fractions?
Yes. You can enter values like 1/2, -3/4, or 5/6. The calculator converts them before row reduction.
7. What is the verification residual?
It measures how close Av is to λv. A very small residual shows the basis vector satisfies the eigenvector equation.
8. Which matrix sizes are supported?
This version supports 2 by 2, 3 by 3, and 4 by 4 square matrices for clear web use.