Eigenvalues and diagonalisation
Lesson 6 of 10
Finding eigenvectors
Objective. Find an eigenvector for a known eigenvalue.
The key idea
Solve (A − λI)v = 0. The solution set is a subspace, so any non-zero multiple of the vector you find is also an eigenvector.
Worked example
For A = [[2, 1], [1, 2]] and λ = 3, find an eigenvector.
- 1A − 3I = [[−1, 1], [1, −1]].
- 2The equation gives −x + y = 0, so y = x.
- 3Take x = 1.
Answer: (1, 1) or any non-zero multiple
Practice
Each question comes with a picture and a listen button. Try it first, then reveal the answer.
Common mistake. Reporting the zero vector as the solution of (A − λI)v = 0.
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