Section 5.3 Eigenvalues and Characteristic Polynomials (G3)
Activity 5.3.1.
An invertible matrix \(M\) and its inverse \(M^{-1}\) are given below:
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 \(M\text{,}\)
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 \(A = \left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right]\text{.}\)
It is easy to see geometrically that
It is less obvious (but easily checked once you find it) that
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 5.3.7.
Finding the eigenvalues \(\lambda\) that satisfy
for some nontrivial eigenvector \(\vec x\) is equivalent to finding nonzero solutions for the matrix 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 5.3.8.
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
Definition 5.3.9.
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
Thus the characteristic polynomial of \(A\) is
and its eigenvalues are the solutions to \(\lambda^2-5\lambda-2=0\text{.}\)
Activity 5.3.10.
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 5.3.11.
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 5.3.1 Videos
Exercises 5.3.2 Exercises
Exercises available at checkit.clontz.org 1 .
https://checkit.clontz.org/#/banks/tbil-la/G3/