Linear Algebra · Quick Reference

Eigenvectors Cheat Sheet

Linear Algebra · Lesson 5/7

In one line: once you know a matrix's eigenvalues, eigenvectors show you the exact directions that only get stretched or shrunk — never rotated — when the matrix acts on them.

Key Ideas

1Eigenvector Definition. A nonzero vector v satisfying A*v = lambda*v for some eigenvalue lambda; the matrix only scales v, never changes its direction.

2Finding Eigenvectors. For each eigenvalue lambda, solve the system (A - lambda*I)*v = 0 to find the corresponding eigenvector(s).

3Eigenvectors Are Not Unique. Any nonzero scalar multiple of an eigenvector is also a valid eigenvector for the same eigenvalue — eigenvectors define a direction, not a single fixed vector.

4Eigenspace. The set of all eigenvectors for a given eigenvalue (plus the zero vector) forms a subspace called the eigenspace for that eigenvalue.

5Diagonalization (Concept). If a matrix has enough independent eigenvectors, it can be rewritten in a simpler diagonal form using those eigenvectors, making many calculations far easier.

Worked Examples

Find the eigenvector of [[4,1],[2,3]] for eigenvalue lambda=5.

Eigenvector = (1,1), or any scalar multiple like (2,2)

Find the eigenvector of [[4,1],[2,3]] for eigenvalue lambda=2.

Eigenvector = (1,-2)

Is (2,2) also a valid eigenvector for the lambda=5 case above (where (1,1) was found)?

Yes, any nonzero scalar multiple of a valid eigenvector is also valid

Formulas

(A - lambda*I) * v = 0

Practice Yourself

For [[3,0],[0,5]] with eigenvalue 3, find an eigenvector.

(1,0)

For [[3,0],[0,5]] with eigenvalue 5, find an eigenvector.

(0,1)

Is (3,3) a valid eigenvector if (1,1) is one?

Yes, it's a scalar multiple (3 times)