Eigenvectors

Japanese school year: University year 1

What you learn

Learn how to find eigenvectors, which are nonzero vectors that become their own scalar multiples scaled by the corresponding eigenvalues when multiplied by the matrix. These vectors are widely applied in principal component analysis and vibration engineering. Knowledge of solving linear systems and finding eigenvalues is required.

Go to practice

Key points

This formula defines an eigenvector $\boldsymbol{v}$ and eigenvalue $\lambda$ . Multiplying vector $\boldsymbol{v}$ by matrix $A$ keeps its direction and only scales it by $\lambda$ ; $\boldsymbol{v}$ cannot be the zero vector.

A\boldsymbol{v} = \lambda\boldsymbol{v} \quad (\boldsymbol{v} \neq \boldsymbol{0})

Use this system of linear equations to find eigenvector $\boldsymbol{v}$ after determining eigenvalue $\lambda$ . Here, $I$ is the identity matrix, and you look for non-zero solutions $\boldsymbol{v}$ .

(A - \lambda I)\boldsymbol{v} = \boldsymbol{0}

This formula defines eigenspace $E_\lambda$ , which collects all eigenvectors for $\lambda$ plus the zero vector. $\operatorname{Ker}$ represents all vectors that become zero when multiplied by $(A - \lambda I)$ .

E_\lambda = \operatorname{Ker}(A - \lambda I)

Choose a set to practice.

Practice 1Coming soon