Learning Outcomes
-
Find the eigenvalues of a \(2\times 2\) matrix.
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.
\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 5.3.3.
For every invertible matrix \(M\text{,}\)
\begin{equation*}
\det(M)\det(M^{-1})= \det(I)=1
\end{equation*}
so \(\det(M^{-1})=\frac{1}{\det(M)}\text{.}\)
Observation 5.3.4.
Consider the linear transformation \(A : \IR^2 \rightarrow \IR^2\) given by the matrix \(A = \left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right]\text{.}\)
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 \(\vec{x}\text{.}\)
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.
What are the eigenvalues for this matrix?
-
\(\displaystyle 1,-2\)
-
\(\displaystyle -1,3\)
-
\(\displaystyle 2,-3\)
-
\(\displaystyle -1,-2\)
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*}
(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 5.3.8.
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 5.3.9.
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 5.3.10.
Let \(A = \left[\begin{array}{cc} 5 & 2 \\ -3 & -2 \end{array}\right]\text{.}\)
(a)
(b)
Set this characteristic polynomial equal to zero and factor to determine the eigenvalues of \(A\text{.}\)
Activity 5.3.11.
Find all the eigenvalues for the matrix \(A=\left[\begin{array}{cc} 3 & -3 \\ 2 & -4 \end{array}\right]\text{.}\)
Activity 5.3.12.
Find all the eigenvalues for the matrix \(A=\left[\begin{array}{cc} 1 & -4 \\ 0 & 5 \end{array}\right]\text{.}\)
Activity 5.3.13.
Find all the eigenvalues for the matrix \(A=\left[\begin{array}{ccc} 3 & -3 & 1 \\ 0 & -4 & 2 \\ 0 & 0 & 7 \end{array}\right]\text{.}\)
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.
(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 5.3.4 Videos
Subsection 5.3.5 Exercises
Exercises available at
https://library.tbil.org/2025/linear-algebra/exercises/#/bank/GT3/
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.Subsection 5.3.7 Sample Problem and Solution
Sample problem Example B.1.24.
