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7.1. Definition of eigenvalue and eigenvector

Definition 7.1.1.

Let $~{}T~{}:~{}{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be a linear transformation.

(i)

An eigenvector of $T$ is a non-zero vector $v\in{\mathbb{R}}^{n}$ such that there exists a scalar $\lambda\in{\mathbb{R}}$ with $T(v)=\lambda v.$
(ii)

The real number $\lambda$ associated to a non-zero eigenvector is called an eigenvalue of $T$ .
(iii)

If $\lambda$ is an eigenvalue of $T$ , the set formed by all vectors $v$ (including zero) such that $Tv=\lambda v$ is called the eigenspace of $T$ corresponding to $\lambda$ . We will denote the eigenspace as $V_{\lambda}$ . Some authors write it as $E(\lambda)$ .

Remark 7.1.2.

Notice that for any linear transformation $T$ , the zero vector satisfies $T(0)=\lambda 0$ , for any real number $\lambda$ . For this reason, we don’t count $0$ as an eigenvector. The reason we want the eigenspace to include zero, is because we want the sum of any two vectors in the eigenspace to be another vector in the eigenspace (in fact, we want it to be a subspace, a concept that will be defined in Section 7.4).

Example 7.1.3.

Let $T=H_{0}:{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ be the linear transformation of reflection about the $x$ -axis. Then the associated matrix is $A=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$ . So every non-zero vector on the $x$ -axis (i.e. any vector of the form $xe_{1}$ ) is mapped to itself by $T$ . In other words, $xe_{1}$ are all eigenvectors with eigenvalue $1$ . On the other hand, every vector $ye_{2}$ is sent to $-ye_{2}$ . In other words, $ye_{2}$ are all eigenvectors for the eigenvalue $-1$ . Summarizing,

$V_{1}=\left\{\begin{pmatrix}x\\ 0\end{pmatrix}\in{\mathbb{R}}^{2}\right\}\quad\hbox{is the eigenspace of $T$ for the eigenvalue $1$, and}\quad$

$V_{-1}=\left\{\begin{pmatrix}0\\ y\end{pmatrix}\in{\mathbb{R}}^{2}\right\}\quad\hbox{is the eigenspace of $T$ for the eigenvalue $-1$.}\quad$

Example 7.1.4.

Consider a linear transformation $T$ whose associated matrix is diagonal (see Definition 4.3.1). Then the entries along the diagonal are the eigenvalues, and the standard basis vectors $e_{i}$ are eigenvectors (but there are other eigenvectors as well).
In particular, for the zero map, every non-zero vector is an eigenvector for the eigenvalue $0$ ; and for the matrix $\lambda I_{n}$ , every non-zero vector is an eigenvector for the eigenvalue $\lambda$ .

Example 7.1.5.

Let $\theta\in[0,2\pi)$ and consider the rotation $R_{\theta}~{}:~{}{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ . Picture a non-zero vector rotating around the origin. For a vector to be an eigenvector of $R_{\theta}$ , it has to be rotated to a scalar multiple of itself. Now you can convince yourself that the only way a rotation by angle $\theta$ sends a vector to a scalar multiple of itself is if $\theta=0$ or $\pi$ . In the case $\theta=0$ , we have $R_{0}=\operatorname{Id}\nolimits$ , so all vectors are eigenvectors with eigenvalue 1. On the other hand, if $\theta=\pi$ , this is rotation by 180 degrees, which is the same as $R_{\pi}=-\operatorname{Id}\nolimits$ ; in this case all vectors are eigenvectors with eigenvalue $-1$ . In particular, the eigenspaces in both cases are the entire plane ${\mathbb{R}}^{2}$ .