4.2 Finding Eigenvalues and Eigenvectors
Reading
Complete the preview activity and read Chapter 4 Section 2 in Understanding Linear Algebra by David Austin.
Do you remember?
Given a matrix [latex]A[/latex], a non-zero vector [latex]\vec{v}[/latex] is an eigenvector of [latex]A[/latex] if [latex]A\vec{v}[/latex] is a multiple of [latex]\vec{v}[/latex]. In other words, [latex]\vec{v}[/latex] is an eigenvector of A if [latex]A\vec{v}= \lambda\vec{v}[/latex]. Note that [latex]\lambda[/latex] can equal zero, but that the zero vector is not considered an eigenvector.
Finding eigenvalues, derivation of the method
Unfortunate typo: video should say “finding eigenvalues”
Examples of finding eigenvalues:
The polynomial [latex]P(\lambda) = det(A-\lambda I)[/latex] is called the characteristic polynomial. If A is an nxn matrix, this polynomial has degree n. Its roots are the eigenvalues of A.
Finding eigenvectors, derivation of the method
Examples of finding eigenvectors:
The eigenvectors of a matrix A for a given eigenvalue [latex]\lambda[/latex] form a subspace (we also throw in [latex]\vec{0}[/latex]), called the eigenspace of A for eigenvalue [latex]\lambda[/latex]. This is written [latex]E_{\lambda} = Nul(A-\lambda I)[/latex].
