Eigenspaces
The set of eigenvectors for an eigenvalue plus zero.
Key idea
The set of eigenvectors for an eigenvalue plus zero.
In depth
The eigenspace for λ is the null space of (A − λI), consisting of all eigenvectors for λ and 0. Its dimension is the geometric multiplicity, which cannot exceed the algebraic multiplicity.
Worked example
For A = I (2 × 2), the eigenspace for 1 is all of R².
Real-world uses
- Finding all stable modes of a system that scale without changing direction
Common mistake
Assuming each eigenvalue has only a single eigenvector.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Eigenspaces?
The set of eigenvectors for an eigenvalue plus zero. It is a topic at the expert level in Linear Algebra.
Where is Eigenspaces used in real life?
Real-world examples: Finding all stable modes of a system that scale without changing direction.
What should I learn before Eigenspaces?
Make sure you are comfortable with Characteristic Polynomial, the previous topic in Linear Algebra.
What is a common mistake with Eigenspaces?
Assuming each eigenvalue has only a single eigenvector.
How can I practise Eigenspaces?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Eigenspaces?
Move on to Diagonalization next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.