## Section5.3Eigenvalues and Characteristic Polynomials (GT3)

### Subsection5.3.1Class Activities

#### Activity5.3.1.

An invertible matrix $$M$$ and its inverse $$M^{-1}$$ are given below:

\begin{equation*} M=\left[\begin{array}{cc}1&2\\3&4\end{array}\right] \hspace{2em} M^{-1}=\left[\begin{array}{cc}-2&1\\3/2&-1/2\end{array}\right] \end{equation*}

Which of the following is equal to $$\det(M)\det(M^{-1})\text{?}$$

1. $$\displaystyle -1$$

2. $$\displaystyle 0$$

3. $$\displaystyle 1$$

4. $$\displaystyle 4$$

#### Observation5.3.3.

Consider the linear transformation $$A : \IR^2 \rightarrow \IR^2$$ given by the matrix $$A = \left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right]\text{.}$$ Figure 65. Transformation of the unit square by the linear transformation $$A$$

It is easy to see geometrically that

\begin{equation*} A\left[\begin{array}{c}1 \\ 0 \end{array}\right] = \left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right]\left[\begin{array}{c}1 \\ 0 \end{array}\right]= \left[\begin{array}{c}2 \\ 0 \end{array}\right]= 2 \left[\begin{array}{c}1 \\ 0 \end{array}\right]\text{.} \end{equation*}

It is less obvious (but easily checked once you find it) that

\begin{equation*} A\left[\begin{array}{c} 2 \\ 1 \end{array}\right] = \left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right]\left[\begin{array}{c}2 \\ 1 \end{array}\right]= \left[\begin{array}{c} 6 \\ 3 \end{array}\right] = 3\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\text{.} \end{equation*}

#### Definition5.3.4.

Let $$A \in M_{n,n}\text{.}$$ An eigenvector for $$A$$ is a vector $$\vec{x} \in \IR^n$$ such that $$A\vec{x}$$ is parallel to $$\vec{x}\text{.}$$ Figure 66. The map $$A$$ stretches out the eigenvector $$\left[\begin{array}{c}2 \\ 1 \end{array}\right]$$ by a factor of $$3$$ (the corresponding eigenvalue).

In other words, $$A\vec{x}=\lambda \vec{x}$$ for some scalar $$\lambda\text{.}$$ If $$\vec x\not=\vec 0\text{,}$$ then we say $$\vec x$$ is a nontrivial eigenvector and we call this $$\lambda$$ an eigenvalue of $$A\text{.}$$

#### Activity5.3.5.

Finding the eigenvalues $$\lambda$$ that satisfy

\begin{equation*} A\vec x=\lambda\vec x=\lambda(I\vec x)=(\lambda I)\vec x \end{equation*}

for some nontrivial eigenvector $$\vec x$$ is equivalent to finding nonzero solutions for the matrix equation

\begin{equation*} (A-\lambda I)\vec x =\vec 0\text{.} \end{equation*}

Which of the following must be true for any eigenvalue?

1. The kernel of the transformation with standard matrix $$A-\lambda I$$ must contain the zero vector, so $$A-\lambda I$$ is invertible.

2. The kernel of the transformation with standard matrix $$A-\lambda I$$ must contain a non-zero vector, so $$A-\lambda I$$ is not invertible.

3. The image of the transformation with standard matrix $$A-\lambda I$$ must contain the zero vector, so $$A-\lambda I$$ is invertible.

4. The image of the transformation with standard matrix $$A-\lambda I$$ must contain a non-zero vector, so $$A-\lambda I$$ is not invertible.

#### Definition5.3.7.

The expression $$\det(A-\lambda I)$$ is called characteristic polynomial of $$A\text{.}$$

For example, when $$A=\left[\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right]\text{,}$$ we have

\begin{equation*} A-\lambda I= \left[\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right]- \left[\begin{array}{cc}\lambda & 0 \\ 0 & \lambda\end{array}\right]= \left[\begin{array}{cc}1-\lambda & 2 \\ 3 & 4-\lambda\end{array}\right]\text{.} \end{equation*}

Thus the characteristic polynomial of $$A$$ is

\begin{equation*} \det\left[\begin{array}{cc}1-\lambda & 2 \\ 3 & 4-\lambda\end{array}\right] = (1-\lambda)(4-\lambda)-(2)(3) = \lambda^2-5\lambda-2 \end{equation*}

and its eigenvalues are the solutions to $$\lambda^2-5\lambda-2=0\text{.}$$

#### Activity5.3.8.

Let $$A = \left[\begin{array}{cc} 5 & 2 \\ -3 & -2 \end{array}\right]\text{.}$$

##### (a)

Compute $$\det (A-\lambda I)$$ to determine the characteristic polynomial of $$A\text{.}$$

##### (b)

Set this characteristic polynomial equal to zero and factor to determine the eigenvalues of $$A\text{.}$$

#### Activity5.3.9.

Find all the eigenvalues for the matrix $$A=\left[\begin{array}{cc} 3 & -3 \\ 2 & -4 \end{array}\right]\text{.}$$

#### Activity5.3.10.

Find all the eigenvalues for the matrix $$A=\left[\begin{array}{cc} 1 & -4 \\ 0 & 5 \end{array}\right]\text{.}$$

#### Activity5.3.11.

Find all the eigenvalues for the matrix $$A=\left[\begin{array}{ccc} 3 & -3 & 1 \\ 0 & -4 & 2 \\ 0 & 0 & 7 \end{array}\right]\text{.}$$

### Subsection5.3.3Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/GT3.slides.html.

### Subsection5.3.5Mathematical Writing Explorations

#### Exploration5.3.12.

What are the maximum and minimum number of eigenvalues associated with an $$n \times n$$ matrix? Write small examples to convince yourself you are correct, and then prove this in generality.

### Subsection5.3.6Sample Problem and Solution

Sample problem Example B.1.23.