Eigenvectors: Directions a Matrix Won't Turn

Lesson notes

Key points

• Most vectors change direction when multiplied by a matrix A, but special ones only get stretched or shrunk.
• The scaling factor lambda is the eigenvalue; lambda of two doubles the vector, negative lambda flips it.
• Rewriting gives (A minus lambda I)v equals zero, which has nonzero solutions only when that matrix is singular.
• So solve the determinant polynomial for each lambda, then solve the homogeneous system to get its eigenvectors.

Definitions

Eigenvector

A nonzero vector v with Av = λv: A only scales v, never turns it off its line.

Takeaway: Eigenvectors keep their direction; eigenvalues tell how much they scale.

EigenvectorEigenvalueCharacteristic equationProcedure