A = Matrix([[3,-2,5],[1,0,7],[0,0,1]])
B = Matrix([[1,0,0,1],[1,2,0,0],[0,0,1,0],[0,0,0,1]])
A.eigenvalues(), B.eigenvalues()
([2, 1, 1], [2, 1, 1, 1])
Math 303: Section 17
\[ \def\R{{\mathbb R}} \def\b{{\mathbf{b}}} \def\x{{\mathbf{x}}} \def\v{{\mathbf{v}}} \def\w{{\mathbf{w}}} \DeclareMathOperator{\null}{Nul} \]
Definition 1 (Eigenvalue and eigenvector) Let \(A\) be an \(n\times n\) matrix. A nonzero vector \(\v\) is an eigenvector of \(A\) if there is a scalar \(\lambda\) such that
\[ A \v = \lambda \v. \]
The scalar \(\lambda\) is called an eigenvalue for \(A\).
How do we find \(\lambda\)?
Definition 2 Let \(A\) be \(n\times n\) matrix. The characteristic polynomial of \(A\) is the polynomial
\[ \det(A-\lambda I_n), \]
where \(I_n\) is the \(n\times n\) identity matrix. The characteristic equation of \(A\) is the equation
\[ \det(A-\lambda I_n) = 0. \]
Solving the characteristic equation gives the eigenvalues of \(A\).
\[ A = \left[\begin{matrix} 3 & -2 & 5 \\ 1 & 0 & 7 \\ 0 & 0 & 1 \end{matrix}\right], \]
and use it to find the eigenvalues of \(A\).
\[ \left[\begin{matrix} 1 & 0 & 0 & 1 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right]. \]
sage
Definition 3 The (algebraic) multiplicity of an eigenvalue \(\lambda\) of a matrix \(A\) is the largest integer \(m\) so that \((x-\lambda)^m\) divides the characteristic polynomial of \(A\).
Theorem 1 (Facts about eigenvalues) Let \(A\) be \(n\times n\) with real entries. Then
For each eigenvalue \(\lambda\) of \(A\), the eigenspace of \(A\) corresponding to the eigenvalue \(\lambda\) is \(\null(A-\lambda I_n)\).
That is, given a matrix \(A\) and eigenvalue \(\lambda\) of \(A\):
Find the dimension of the eigenspace for each eigenvalue of \(A = \left[\begin{matrix} 3 & -2 & 5 \\ 1 & 0 & 7 \\ 0 & 0 & 1 \end{matrix}\right]\).
Find the dimension of the eigenspace for each eigenvalue of \(A = \left[\begin{matrix} 1 & 0 & 0 & 1 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right]\).
Consider a \(3\times 3\) matrix with three distinct eigenvalues \(\lambda_1, \lambda_2, \lambda_3\).
Definition 4 (Geometric Multiplicity) The geometric multiplicity of an eigenvalue of an \(n\times n\) matrix \(A\) is the dimension of the corresponding eigenspace \(\null(A-\lambda I_n)\).
Exercise 1 Find the eigenvalues of the following matrices. For each eigenvalue, determine its algebraic and geometric multiplicity.
\(A = \left[\begin{matrix} 5 & -8 & 1 \\ 0 & 0 & 7 \\ 0 & 0 & -2 \end{matrix}\right]\)
\(B = \left[\begin{matrix} 2 & 4 & 3 \\ -4 & -6 & -3 \\ 3 & 3 & 1 \end{matrix}\right]\)