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Section 5.3 Eigenvalues and Characteristic Polynomials (GT3)

Subsection 5.3.1 Warm Up

Activity 5.3.1.

Let \(R\colon\IR^2\to\IR^2\) be the transformation given by rotating vectors about the origin through and angle of \(45^\circ\text{,}\) and let \(S\colon\IR^2\to\IR^2\) denote the transformation that reflects vectors about the line \(x_1=x_2\text{.}\)

(a)

If \(L\) is a line, let \(R(L)\) denote the line obtained by applying \(R\) to it. Are there any lines \(L\) for which \(R(L)\) is parallel to \(L\text{?}\)

(b)

Now consider the transformation \(S\text{.}\) Are there any lines \(L\) for which \(S(L)\) is parallel to \(L\text{?}\)

Subsection 5.3.2 Class Activities

Activity 5.3.2.

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. \(\displaystyle 4\)

Observation 5.3.4.

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 5.3.5.

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 5.3.6.

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*}

(a)

If \(\lambda\) is an eigenvalue, and \(T\) is the transformation with standard matrix \(A-\lambda I\text{,}\) which of these must contain a non-zero vector?
  1. The kernel of \(T\)
  2. The image of \(T\)
  3. The domain of \(T\)
  4. The codomain of \(T\)

Definition 5.3.8.

The expression \(\det(A-\lambda I)\) is called the characteristic polynomial of \(A\text{.}\)
For example, when \(A=\left[\begin{array}{cc}1 & 2 \\ 5 & 4\end{array}\right]\text{,}\) we have
\begin{equation*} A-\lambda I= \left[\begin{array}{cc}1 & 2 \\ 5 & 4\end{array}\right]- \left[\begin{array}{cc}\lambda & 0 \\ 0 & \lambda\end{array}\right]= \left[\begin{array}{cc}1-\lambda & 2 \\ 5 & 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 \\ 5 & 4-\lambda\end{array}\right] = (1-\lambda)(4-\lambda)-(2)(5) = \lambda^2-5\lambda-6 \end{equation*}
and its eigenvalues are the solutions \(-1,6\) to \(\lambda^2-5\lambda-6=0\text{.}\)

Activity 5.3.9.

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 5.3.10.

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

Activity 5.3.11.

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

Activity 5.3.12.

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 5.3.3 Cool Down

Activity 5.3.13.

Let \(A\in M_{n,n}\) and \(\lambda\in\IR\text{.}\) The eigenvalues of \(A\) that correspond to \(\lambda\) are the vectors that get stretched by a factor of \(\lambda\text{.}\) Consider the following special cases for which we can make more geometric meaning.

(a)

What are some other ways we can think of the eigenvalues corresponding to eigenvalue \(\lambda=0\text{?}\)

(b)

What are some other ways we can think of the eigenvalues corresponding to eigenvalue \(\lambda=1\text{?}\)

(c)

What are some other ways we can think of the eigenvalues corresponding to eigenvalue \(\lambda=-1\text{?}\)

(d)

How might we interpret a matrix that has no (real) eigenvectors/values?

Subsection 5.3.4 Videos

Figure 67. Video: Finding eigenvalues

Exercises 5.3.5 Exercises

Subsection 5.3.6 Mathematical Writing Explorations

Exploration 5.3.14.

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 5.3.7 Sample Problem and Solution

Sample problem Example B.1.24.
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