TBIL-LA Eigenvalues and Characteristic Polynomials (GT3)

Section 5.3 Eigenvalues and Characteristic Polynomials (GT3)

Learning Outcomes

Find the eigenvalues of a \(2\times 2\) matrix.

Subsection 5.3.1 Class Activities

Activity 5.3.1.

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 \(\det(M)\det(M^{-1})\text{?}\)

\(\displaystyle -1\)
\(\displaystyle 0\)
\(\displaystyle 1\)
\(\displaystyle 4\)

Fact 5.3.2.

For every invertible matrix \(M\text{,}\)

\begin{equation*} \det(M)\det(M^{-1})= \det(I)=1 \end{equation*}

so \(\det(M^{-1})=\frac{1}{\det(M)}\text{.}\)

Furthermore, a square matrix \(M\) is invertible if and only if \(\det(M)\not=0\text{.}\)

Observation 5.3.3.

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*}

Definition 5.3.4.

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 \(A\text{.}\)

Activity 5.3.5.

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?

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

Fact 5.3.6.

The eigenvalues \(\lambda\) for a matrix \(A\) are the values that make \(A-\lambda I\) non-invertible.

Thus the eigenvalues \(\lambda\) for a matrix \(A\) are the solutions to the equation

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

Definition 5.3.7.

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 \(\lambda^2-5\lambda-2=0\text{.}\)

Activity 5.3.8.

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 \(A\text{.}\)

Activity 5.3.9.

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

Activity 5.3.10.

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

Activity 5.3.11.

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

Subsection 5.3.2 Videos

Figure 67. Video: Finding eigenvalues

Subsection 5.3.3 Slideshow

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

Exercises 5.3.4 Exercises

Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/GT3/.

Subsection 5.3.5 Mathematical Writing Explorations

Exploration 5.3.12.

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.

Subsection 5.3.6 Sample Problem and Solution

Sample problem Example B.1.23.

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