Learning Outcomes
- Find the eigenvalues of a \(2\times 2\) matrix.
Section 22.3 Eigenvalues and Characteristic Polynomials (GT3)
Subsection 22.3.1 Warm Up
Activity 22.52.
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 22.3.2 Class Activities
Activity 22.53.
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 22.54.
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 22.55.
Consider the linear transformation \(A : \IR^2 \rightarrow \IR^2\) given by the matrix \(A = \left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right]\text{.}\)
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 22.56.
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 \(\vec{x}\text{.}\)
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 22.57.
What are the eigenvalues for this matrix?
- \(\displaystyle 1,-2\)
- \(\displaystyle -1,3\)
- \(\displaystyle 2,-3\)
- \(\displaystyle -1,-2\)
Activity 22.58.
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?
- The kernel of \(T\)
- The image of \(T\)
- The domain of \(T\)
- The codomain of \(T\)
(b)
Therefore, what can we conclude?
- \(A\) is invertible
- \(A\) is not invertible
- \(A-\lambda I\) is invertible
- \(A-\lambda I\) is not invertible
(c)
And what else?
- \(\displaystyle \det A=0\)
- \(\displaystyle \det A=1\)
- \(\displaystyle \det(A-\lambda I)=0\)
- \(\displaystyle \det(A-\lambda I)=1\)
Fact 22.59.
The eigenvalues \(\lambda\) for a matrix \(A\) are exactly 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 22.60.
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 22.61.
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 22.62.
Find all the eigenvalues for the matrix \(A=\left[\begin{array}{cc} 3 & -3 \\ 2 & -4 \end{array}\right]\text{.}\)
Activity 22.63.
Find all the eigenvalues for the matrix \(A=\left[\begin{array}{cc} 1 & -4 \\ 0 & 5 \end{array}\right]\text{.}\)
Activity 22.64.
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 22.3.3 Individual Practice
Activity 22.65.
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 eigenvectors corresponding to eigenvalue \(\lambda=0\text{?}\)
(b)
What are some other ways we can think of the eigenvectors corresponding to eigenvalue \(\lambda=1\text{?}\)
(c)
What are some other ways we can think of the eigenvectors corresponding to eigenvalue \(\lambda=-1\text{?}\)
(d)
How might we interpret a matrix that has no (real) eigenvectors/values?
Subsection 22.3.4 Videos
Exercises 22.3.5 Exercises
Exercises available at
https://tbil.org/preview/linear-algebra/exercises/#/bank/GT3/.Subsection 22.3.6 Mathematical Writing Explorations
Exploration 22.66.
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 22.3.7 Sample Problem and Solution
Sample problem Example C.24.
