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Linear Algebra for Team-Based Inquiry Learning PREVIEW Edition

Steven Clontz, Drew Lewis

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 \(x_1=x_2\text{.}\)
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(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 \(L\text{?}\)
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(b)
Now consider the transformation \(S\text{.}\) Are there any lines \(L\) for which \(S(L)\) is parallel to \(L\text{?}\)
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Subsection 5.3.2 Class Activities

Activity 5.3.2.

An invertible matrix \(M\) and its inverse \(M^{-1}\) are given below:
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\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 \(\det(M)\det(M^{-1})\text{?}\)
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  1. \(\displaystyle -1\)
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  2. \(\displaystyle 0\)
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  3. \(\displaystyle 1\)
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  4. \(\displaystyle 4\)
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Observation 5.3.4.

Consider the linear transformation \(A : \IR^2 \rightarrow \IR^2\) given by the matrix \(A = \left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right]\text{.}\)
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described in detail following the image
A figure in the \(xy\)-plane. A blue vector pointing right along the \(x\)-axis is labeled \(A\vec{e}_1\text{.}\) A blue vector pointing up and right is labeled \(A\vec{e}_2\text{.}\) Dotted blue lines complete the parallelogram spanned by these two vectors. A red horizontal vector labeled \(\vec{e}_1\) and a red vertical vector labeled \(\vec{e}_2\) form the edges of a red square. A red vector points up and to the right through the middle of the parallelogram, and a purple, longer vector parallel to this vector extends out further.
Figure 59. Transformation of the unit square by the linear transformation \(A\)
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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*}
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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*}
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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 \(\vec{x}\text{.}\)
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described in detail following the image
The image from Figure 59 is reproduced with most of the parts grayed out. The red vector pointing up and to the right is now labeled \(\left[\begin{array}{c}2 \\ 1 \end{array}\right]\text{,}\) and the purple parallel vector remains; it is labeled to the right with \(A\left[\begin{array}{c}2 \\ 1\end{array}\right]=3\left[\begin{array}{c}2 \\ 1\end{array}\right]\text{.}\) The red vector \(\vec{e}_1\) remains, and the parallel blue vector is now labeled \(A\vec{e}_1=2\vec{e}_1\text{.}\)
Figure 60. The map \(A\) stretches out the eigenvector \(\left[\begin{array}{c}2 \\ 1 \end{array}\right]\) by a factor of \(3\) (the corresponding eigenvalue).
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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 \(A\text{.}\)
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Activity 5.3.6.

What are the eigenvalues for this matrix?
  1. \(\displaystyle 1,-2\)
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  2. \(\displaystyle -1,3\)
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  3. \(\displaystyle 2,-3\)
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  4. \(\displaystyle -1,-2\)
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Activity 5.3.7.

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*}
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(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?
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  1. The kernel of \(T\)
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  2. The image of \(T\)
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  3. The domain of \(T\)
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  4. The codomain of \(T\)
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Definition 5.3.9.

The expression \(\det(A-\lambda I)\) is called the characteristic polynomial of \(A\text{.}\)
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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*}
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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 \(\lambda^2-5\lambda-6=0\text{.}\)
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Activity 5.3.10.

Let \(A = \left[\begin{array}{cc} 5 & 2 \\ -3 & -2 \end{array}\right]\text{.}\)
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(a)
Compute \(\det (A-\lambda I)\) to determine the characteristic polynomial of \(A\text{.}\)
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(b)
Set this characteristic polynomial equal to zero and factor to determine the eigenvalues of \(A\text{.}\)
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Activity 5.3.11.

Find all the eigenvalues for the matrix \(A=\left[\begin{array}{cc} 3 & -3 \\ 2 & -4 \end{array}\right]\text{.}\)
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Activity 5.3.12.

Find all the eigenvalues for the matrix \(A=\left[\begin{array}{cc} 1 & -4 \\ 0 & 5 \end{array}\right]\text{.}\)
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Activity 5.3.13.

Find all the eigenvalues for the matrix \(A=\left[\begin{array}{ccc} 3 & -3 & 1 \\ 0 & -4 & 2 \\ 0 & 0 & 7 \end{array}\right]\text{.}\)
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Subsection 5.3.3 Individual Practice

Activity 5.3.14.

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.
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(a)
What are some other ways we can think of the eigenvectors corresponding to eigenvalue \(\lambda=0\text{?}\)
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(b)
What are some other ways we can think of the eigenvectors corresponding to eigenvalue \(\lambda=1\text{?}\)
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(c)
What are some other ways we can think of the eigenvectors corresponding to eigenvalue \(\lambda=-1\text{?}\)
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(d)
How might we interpret a matrix that has no (real) eigenvectors/values?
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Subsection 5.3.4 Videos

Figure 61. Video: Finding eigenvalues
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Subsection 5.3.5 Exercises

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Subsection 5.3.6 Mathematical Writing Explorations

Exploration 5.3.15.

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.
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Subsection 5.3.7 Sample Problem and Solution

Sample problem Example B.1.24.
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