6.5 Eigenspaces - ESE 2030 📏

Lecture notes

1Reading¶

Material related to this page, as well as additional exercises, can be found in ALA 8.1.

2Learning Objectives¶

By the end of this page, you should know:

how to define the eigenspace corresponding to an eigenvalue.

3Eigenspaces¶

Before we introduce the notion of an eigenspace, let’s first consider a motivating example:

Example 1 (A characteristic equation with a double root)

Consider the matrix $A = \bm 2&-1&-1\\0&3&1\\0&1&3 \em$ . To find its eigenvalues, we can solve:

\begin{align*} \det\bm 2-\lambda&-1&-1\\0&3-\lambda&1\\0&1&3-\lambda \em = 0 \end{align*}

(1)

Using determinant Fact 4,

\begin{align*} \det\bm 2-\lambda&-1&-1\\0&3-\lambda&1\\0&1&3-\lambda \em &= \det (2 - \lambda) \det \bm 3-\lambda&1\\1&3-\lambda \em \\ &= (2 - \lambda)(\lambda - 4)(\lambda - 2) \end{align*}

(2)

Most $3 \times 3$ matrices have distinct eigenvalues, but this one only has two. The eigenvalue $\lambda_1 = 2$ is a double eigenvalue, because it is a double root of the characteristic equation, while the eigenvalue $\lambda_2 = 4$ is a simple eigenvalue.

The eigenvector equation for $\lambda_1 = 2$ is:

\begin{align*} &\bm 2-\lambda&-1&-1\\0&3-\lambda&1\\0&1&3-\lambda \em - 2I = \bm 0&-1&-1\\0&1&1\\0&1&1 \em = 0 \end{align*}

(3)

which has general solution given by $\text{span}\left\{ \bm 1\\0\\0 \em, \bm 0\\1\\-1 \em \right\}$ . This is the eigenspace corresponding to an eigenvalue of 2.

Similarly, the general solution to the eigenvector equation for $\lambda_2 = 4$ is $\text{span}\left\{ \bm -1\\1\\1 \em \right\}$ . This is the eigenspace corresponding to an eigenvalue of 4.

In summary, the eigenvalues and “basis eigenvector” pairs for this matrix are:

\begin{align*} \lambda_1 = 2,&\quad \vv{v_1} = \bm 1\\0\\0 \em, \vv{\hat{v_1}} = \bm 0\\1\\-1 \em\\ \lambda_2 = 4,& \quad \vv{v_2} = \bm -1\\1\\1 \em. \end{align*}

(4)

In general, we can define the eigenspace of an eigenvalue λ as follows:

This description also gives us a very important connection between zero eigenvalues and the invertibility of a matrix $A$ :

Ch 6 Eigenvalues And Eigenvectors

6.4 The QR Algorithm for Eigenvalues

Ch 6 Eigenvalues And Eigenvectors

6.6 Repeated and Complex Eigenvalues