4.1 An Introduction to eigenvalues and eigenvectors
Readings
At this time please read Chapter 4 Section 1 in Understanding Linear Algebra by David Austin.
Activity
Please complete the Stretch Factors and Directions Worksheet.
Let A be a matrix. If there is a non-zero vector [latex]\vec{v}[/latex] such that [latex]A\vec{v} = \lambda \vec{v}[/latex], for some scalar [latex]\lambda[/latex] we call [latex]\vec{v}[/latex] an eigenvector and [latex]\lambda[/latex] its eigenvalue. As [latex]A\vec{0} = \vec{0}[/latex] for all matrices [latex]A[/latex], we do NOT consider [latex]\vec{0}[/latex] to be an eigenvector.
Eigenvector Facts
- All elements of [latex]Nul(A) = \{\vec{x} | A \vec{x} = \vec{0}\}[/latex] except [latex]\vec{0}[/latex] are eigenvectors of [latex]A[/latex]. Their associated eigenvector is 0. Justify!
- If [latex]\vec{v}[/latex] is an eigenvector of matrix A with eigenvalue [latex]\lambda[/latex], then [latex]k \vec{v}[/latex] is also an eigenvector with the same eigenvalue for every [latex]k \neq 0[/latex]. Justify!
- If [latex]\vec{v}[/latex] is an eigenvector of matrix A with eigenvalue [latex]\lambda[/latex], then [latex]\vec{v}[/latex] is an eigenvector for [latex]A^n,[/latex] with eigenvalue [latex]\lambda^n,[/latex] for n=1,2,3,… Check!