Example 8.1.1.
Find the eigenvalues and eigenvectors of the matrix
\begin{equation*}
A = \begin{pmatrix}
2 \amp -12\\
1 \amp -5
\end{pmatrix}
\end{equation*}
Section 8.1 Eigenvalue and Eigenvector
Subsection 8.1.1 Eigenvalue of a square matrix
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, \((A-\lambda I_n) \mathbf{x}=\mathbf{0}\text{.}\)
The matrix equation \(A\mathbf{x}=\mathbf{0}\) has nontrivial solution if and only if \(\underline{\operatorname{det}(A)= 0}\)
Thus the eigenvalue of a matrix should satisfy \(\underline{\hspace{5cm}}\text{.}\) The corresponding eigenvectors should satisfy \(\underline{\hspace{5cm}}\text{.}\)
Exercise: Find the eigenvalues 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*}
Subsection 8.1.2 Characteristic Polynomial
Characteristic Polynomial.
Let \(A\) be a square matrix. The polynomial \(\det(\lambda I_n-A)\) is called the characteristic polynomial of \(A\text{.}\)
Example: Find the characteristic polynomial of the matrix \(A\) and \(B\) above, respectively.
Subsection 8.1.3 Algebraic and Geometric Multiplicities
Let \(f(\lambda)=\det(A-\lambda I_n)\) be the characteristic polynomial of \(A\text{.}\) Suppose that
\begin{align*}
\operatorname{det}(A-\lambda I) \amp =(-1)^{n}\lambda^n+a_{n-1}\lambda^{n-1}+\ldots+a_1\lambda+a_0 \\
\amp =(\lambda_{i_1}-\lambda)(\lambda_{i_2}-\lambda)\ldots (\lambda_{i_n}-\lambda)\\
\amp = (\lambda_1-\lambda)^{m_1}(\lambda_2-\lambda)^{m_2}\ldots (\lambda_k-\lambda)^{m_k}
\end{align*}
Definition 8.1.2.
- The number \(k_{i}\) is called the algebriac multiplicity of the eigenvalue \(\lambda_i\text{.}\)
- 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 \(\{\mathbf{x}\in \mathbb{R}^{n}|(A-\lambda_{i}I_{n})\mathbf{x}=\mathbf{0}\}.\)
Note that the sum of algebraic multiplicities of all eigenvalues is the size of the matrix \(A\text{.}\) Thus if for every eigenvalue \(\lambda_{i}\text{,}\) the geometric multiplicity is equal to the algebraic multiplicity, then we will have \(n\) linear independent eigenvectors.
Theorem 8.1.3.
If for every eigenvalue \(\lambda\) of \(A\text{,}\) the algebraic multiplicity equals the geometric multiplicity, then the matrix \(A\) is diagonalizable.
