Eigenvalues and Eigenvectors

by Qiang Gao, Mar 20, 2017

Definition (eigenvalues and eigenvectors)

For any square matrix $\mathbf{A}$, if a nonzero vector $\mathbf{x}$ satisfies

$$\mathbf{A} \mathbf{x} = \lambda \mathbf{x}$$

for some scalar $\lambda$, then $\lambda$ is called an eigenvalue and $\mathbf{x}$ is called an eigenvector corresponding to it.

Note

• Intuitively, eigenvectors are special vectors that do not change direction after being hit by matrix $\mathbf{A}$. That is, $\mathbf{A}$ carries eigenvector $\mathbf{x}$ to $\lambda \mathbf{x}$, the same direction as $\mathbf{x}$, scaling by a factor of the eigenvalue $\lambda$.

• The eigenvalue can be a complex number (see characteristic equation). If this is true, the corresponding eigenvector must be a complex vector.

• Eigenvalues and eigenvectors are defined simultaneously. That is, if we say $\mathbf{x}$ is an eigenvector of square matrix $\mathbf{A}$, then there must be a corresponding eigenvalue $\lambda$. Likewise, if we say $\lambda$ is an eigenvalue of square matrix $\mathbf{A}$, then there must be a corresponding eigenvector $\mathbf{x}$. In short, they come in pairs.

Theorem 1 (eigenvector means a direction)

If $\mathbf{x}$ is an eigenvector of square matrix $\mathbf{A}$ corresponding to eigenvalue $\lambda$, then $c \mathbf{x}$ is also an eigenvector of square matrix $\mathbf{A}$ corresponding to the same eigenvalue $\lambda$.

Proof

By definition,

$$\mathbf{A} \mathbf{x} = \lambda \mathbf{x},$$

$$c \mathbf{A} \mathbf{x} = c \lambda \mathbf{x},$$

$$\mathbf{A} (c \mathbf{x}) = \lambda (c \mathbf{x}).$$

So $c \mathbf{x}$ is also an eigenvector, by definition.

Note

This implies that the eigenvector means a direction, where its length is irrelevant. As a result of this theorem, it is suffice for us to use only eigenvectors of unit length.

Copyright ©2017 by Qiang Gao