Objectives: Topics
-
Eigenvectors, eigenvalues, and eigenspaces.
-
Eigenvalue theorems.
Print preview
Section 5.1 Eigenvectors, Eigenvalues, and Eigenspaces
Handout 5.1 Eigenvectors, Eigenvalues, and Eigenspaces
Objectives: Goals
-
Verify that a given vector is an eigenvector of a matrix.
-
Verify that a given scalar is an eigenvalue of a matrix.
-
Construct the eigenspace for a given eigenvalue of a matrix.
-
Apply eigenvalue theorems (e.g., to characterize invertibility).
Definition 5.1. Eigenvector and Eigenvalue.
Let \(A\in\mathbb{R}^{n\times n}\text{.}\) If there exists a nonzero vector \(\vec v\in\mathbb{R}^n\) and a scalar \(\lambda\in\mathbb{C}\) such that
\begin{equation*}
A\vec v = \lambda \vec v,
\end{equation*}
Notes:
-
We consider only square matrices.
-
If \(\lambda\in\mathbb{R}\text{,}\) then:
-
for \(\lambda>0\text{,}\) vectors \(A\vec v\) and \(\vec v\) point in the same direction;
-
for \(\lambda<0\text{,}\) they point in opposite directions.
-
-
Even when entries of \(A\) and \(\vec v\) are real, \(\lambda\) can be complex (e.g., a rotation by \(90^\circ\) counterclockwise in \(\mathbb{R}^2\) has no real eigenvalues).
-
Complex eigenvalues will be explored in Section 5.5.
Example 5.2.
Which of the following are eigenvectors of
\begin{equation*}
A=\begin{bmatrix}
1 \amp 1\\
1 \amp 1
\end{bmatrix}?
\end{equation*}
What are the corresponding eigenvalues?
(a)
\(\vec v_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}\)
(b)
\(\vec v_2 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}\)
(c)
\(\vec v_3 = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\)
Example 5.3.
Confirm that \(\lambda=3\) is an eigenvalue of
\begin{equation*}
A=\begin{bmatrix}
2 \amp -4\\
-1 \amp -1
\end{bmatrix}.
\end{equation*}
Definition 5.4. Eigenspace.
Suppose \(A\in\mathbb{R}^{n\times n}\text{.}\) For an eigenvalue \(\lambda\) of \(A\text{,}\) the set of all eigenvectors associated with \(\lambda\) together with the zero vector forms a subspace of \(\mathbb{R}^n\) called the \(\lambda\)-eigenspace of \(A\text{.}\)
Example 5.5.
Construct a basis for the eigenspaces of
\begin{equation*}
A=\begin{bmatrix}
5 \amp -6\\
3 \amp -4
\end{bmatrix},
\qquad \lambda=-1,\;2.
\end{equation*}
Theorem 5.6. Eigenvalues of Triangular Matrices.
The eigenvalues of a triangular (upper or lower) matrix are its diagonal entries.
Theorem 5.7. Invertibility and the Eigenvalue 0.
Theorem 5.8. Stochastic Matrices.
A (column-)stochastic matrix always has an eigenvalue equal to \(1\text{.}\)
Theorem 5.9. Distinct Eigenvalues Give Independent Eigenvectors.
If \(\vec v_1,\vec v_2,\ldots,\vec v_k\) are eigenvectors corresponding to distinct eigenvalues of \(A\text{,}\) then \(\{\vec v_1,\ldots,\vec v_k\}\) is linearly independent.
Note 5.10. Warning.
You cannot determine the eigenvalues of a matrix from its reduced row-echelon form. Row operations generally change eigenvalues.
Example 5.11.
Let
\begin{equation*}
A=\begin{bmatrix}
1 \amp 1\\
1 \amp 1
\end{bmatrix},
\end{equation*}
which has eigenvalues \(\lambda_1=2\) and \(\lambda_2=0\text{.}\)
(a)
Find eigenvectors for each eigenvalue of \(A\text{.}\)
(b)
Find the RREF of \(A\text{.}\) What are it’s eigenvalues?
Note 5.12. Converse of Theorem 5.9.
The converse of Theorem 5.9 is the following:
If \(\{\vec v_1, \vec v_2, \dots, \vec v_k\}\) are linearly independent eigenvectors of \(A\text{,}\) then \(\vec v_1, \vec v_2, \dots, \vec v_k\) must correspond to different eigenvalues.
This statement is false. See Example 5.13 below.
Example 5.13.
Let
\begin{equation*}
A = \begin{bmatrix} 4 \amp -1 \amp 6 \\ 2 \amp 1 \amp 6 \\ 2 \amp -1 \amp 8 \end{bmatrix}.
\end{equation*}
One of the eigenvalues of \(A\) is \(\lambda = 2\text{.}\) Find a basis for the \(2\)-eigenspace of \(A\text{.}\)
