Find the eigenvalues and eigenvectors of the matrix
\begin{equation*}
A = \begin{pmatrix}
2 \amp -12\\
1 \amp -5
\end{pmatrix}
\end{equation*}
Section 7.1 Eigenvalue and Eigenvector
Subsection 7.1.1 Eigenvalue of a square matrix
Note that the matrix equation \(A\mathbf{x}=\mathbf{0}\) has nontrivial solution if and only if \(\underline{\operatorname{det}(A)= 0}\)
Let \(A\) be an \(n \times n\) matrix. The scalar \(\lambda\) is called an eigenvalue of \(A\) when there is a nonzero vector \(\mathbf{x}\) such that
\begin{equation*}
A \mathbf{x}=\lambda \mathbf{x}.
\end{equation*}
The vector \(\mathbf{x}\) is an eigenvector of \(A\) corresponding to \(\lambda\text{.}\)
To solve the matrix equation above, the key is to notice that \(\lambda \mathbf{x}=(\lambda I_n)\mathbf{x} \text{.}\) Thus \(A \mathbf{x}=\lambda \mathbf{x}\) is equivalent to \(A \mathbf{x}-(\lambda I_n) \mathbf{x}=\mathbf{0}\) that is, \((A-\lambda I_n) \mathbf{x}=\mathbf{0}\text{.}\)
Thus the eigenvalue of a matrix should satisfy \(\underline{\hspace{5cm}}\text{.}\) The corresponding eigenvectors should satisfy \(\underline{\hspace{5cm}}\text{.}\)
Example 7.1.1.
Exercise: Find the eigenvalues and the corresponding eigenvectors of the matrix
\begin{equation*}
B = \begin{pmatrix}
1 \amp 2 \amp -2 \\
-2 \amp 5 \amp -2 \\
-6 \amp 6 \amp -3
\end{pmatrix}\text{.}
\end{equation*}
How to find eigenvalues and eigenvectors of a matrix.
To find the eigenvalues and eigenvectors of a matrix \(A\) in SageMath, you can follow these steps:
-
To find the eigenvalues of \(A\text{,}\) use
A.eigenvalues(). -
To find the eigenvectors of \(A\text{,}\) use
A.eigenvectors_right().
Subsection 7.1.2 Characteristic Polynomial
Characteristic Polynomial.
Let \(A\) be a square matrix. The polynomial \(\det(A-\lambda I_n)\) is called the characteristic polynomial of \(A\text{.}\) The eigenvalues of \(A\) are the roots of the characteristic equation \(\det(A-\lambda I_n)=0\) .
Activity 7.1.1.
Use the matrix \(B\) above to verify the following two important facts.
-
The sum of eigenvalues of \(B\) is equal to the trace of \(B\text{,}\) that is,\begin{equation*} \lambda_1+\lambda_2+\ldots+\lambda_n=\operatorname{trace}(B). \end{equation*}
-
The product of eigenvalues of \(B\) is equal to the determinant of \(B\text{,}\) that is,\begin{equation*} \lambda_1\lambda_2\ldots\lambda_n=\operatorname{det}(B). \end{equation*}
Subsection 7.1.3 Algebraic and Geometric Multiplicities
Definition 7.1.2.
-
The Nullity(\(A-\lambda_{i}I_{n}\)) is called the geometric multiplicity of the eigenvalue \(\lambda_{i}\text{.}\) Note that the Nullity is always \(\geq 1\text{.}\) Recall that the nullity is the dimension of the subspace \(\{\mathbf{x}\in \mathbb{R}^{n}|(A-\lambda_{i}I_{n})\mathbf{x}=\mathbf{0}\}.\)
The algebraic multiplicity and geometric multiplicity of an eigenvalue can be read directly from
A.eigenvectors_right().
Exercises Exercises
1.
Find the characteristic polynomial, eigenvalues, and the corresponding eigenvectors of the matrix
\begin{equation*}
B = \begin{pmatrix} 2 \amp 1 \amp 0 \amp 0 \\ 0 \amp 2 \amp 1 \amp 0 \\ 0 \amp 0 \amp 3 \amp 0 \\ 0 \amp 1 \amp -1 \amp 3 \end{pmatrix}\text{.}
\end{equation*}
Determine the algebraic and geometric multiplicities of each eigenvalue.
Subsection 7.1.4 The eigenvalue and eigenvectors of \(f(A)\)
Let \(A\) be an \(n\times n\) matrix with eigenpairs \((\lambda_i,\mathbf{v}_i)\text{,}\) \(i=1,\dots,k\text{.}\) For a polynomial function \(f\) we define
\begin{equation*}
f(A)=a_0 I + a_1A + a_2A^2 + \cdots +a_n A^n.
\end{equation*}
Theorem 7.1.3.
If \(A\mathbf{v}=\lambda\mathbf{v}\) with \(\mathbf{v}\neq\mathbf{0}\) and \(f\) is a polynomial, then
\begin{equation*}
f(A)\mathbf{v}=f(\lambda)\mathbf{v}.
\end{equation*}
Hence \(\lambda\) maps to \(f(\lambda)\) while \(\mathbf{v}\) remains an eigenvector.
Subsubsection 7.1.4.1 Spectral mapping and examples
Example 7.1.4. Example.
Let matrix \(A\) be an \(n \times n\) square matrix. Let \(\mathbf{v}\) be an eigenvector of \(A\) corresponding to the eigenvalue \(\lambda\text{.}\) Prove that:
(b) \(\mathbf{v}\) is an eigenvector of \(A^3 + A^2 + I\text{,}\) and find the corresponding eigenvalue.
