Eigenvector and eigenvalue
An eigenvector of A is a non-zero vector v with A v = lambda v. The matrix only stretches it, and the stretch factor lambda is the eigenvalue.
Last updated
Analogy
Pulling a rubber sheet: some lines of fabric only get longer or shorter and do not turn.
Example
A v = (4+1, 2+3) = (5,5).That equals 5 x (1,1).So v is an eigenvector.Its eigenvalue is 5.
In practice
Eigenvectors reveal the main directions of a transformation and make hard problems simple.
Practice
Flashcards
- What equation defines an eigenvector?A v = lambda v, with v not zero.
Questions people ask
Can the zero vector be an eigenvector?
No. The zero vector satisfies A v equals lambda v for every lambda, so it would tell you nothing. Eigenvectors must be non-zero. The eigenvalue itself can be zero, though, which happens when the matrix squashes that direction to nothing.
Does an eigenvector keep its direction?
Yes, up to a flip. A positive eigenvalue stretches it along the same line, while a negative eigenvalue also reverses it. A matrix with eigenvalue -2 turns the vector into twice its length, pointing the opposite way along the same line.