1. What is an Eigenspace?

The eigenspace for an eigenvalue λ of a matrix A is the null space of (A - λI), where I is the identity matrix. It consists of all vectors v such that Av = λv.

2. How Does the Calculator Work?

The calculator uses the definition:

Where:

• \( A \) — Square matrix
• \( \lambda \) — Eigenvalue of A
• \( I \) — Identity matrix of same size as A
• \( v \) — Eigenvector corresponding to λ

Explanation: The calculator finds all vectors v that satisfy the equation (A - λI)v = 0, which form a vector space called the eigenspace.

3. Importance of Eigenspace Calculation

Details: Eigenspaces are fundamental in linear algebra, used in diagonalization, stability analysis, and understanding linear transformations.

4. Using the Calculator

Tips: Enter a square matrix with rows separated by newlines and columns by spaces. Enter a valid eigenvalue. The calculator will return a basis for the eigenspace and its dimension.

5. Frequently Asked Questions (FAQ)

Q1: What if my matrix isn't square?
A: The calculator only works for square matrices. Non-square matrices don't have eigenvalues/eigenspaces in the traditional sense.

Q2: What does an eigenspace dimension of 0 mean?
A: It means the only solution is the zero vector, indicating λ is not actually an eigenvalue of A.

Q3: How is the basis calculated?
A: The calculator uses Gaussian elimination to find the null space of (A - λI).

Q4: What's the relationship between eigenspace and geometric multiplicity?
A: The dimension of the eigenspace is the geometric multiplicity of the eigenvalue.

Q5: Can I use complex eigenvalues?
A: This calculator only handles real eigenvalues. For complex eigenvalues, you'd need a more advanced calculator.