In the previous section, we saw how advantageous our use of eigenvalues and eigenvectors was to describing the long term behavior of the linear discrete dynamical systems. We will take a few minutes here to set up the algebra of what was going on more fully.
Subsection3.2.1Same Eigenvalues, Different Eigenvectors
Activity3.2.1.
(a)
Find the eigenvalues for the matrices \(A=\begin{bmatrix}2 \amp 0 \\0 \amp 1/2\end{bmatrix}\) and \(B=\begin{bmatrix}1.25 \amp -0.75 \\-0.75 \amp 1.25\end{bmatrix}\text{.}\) For each of the eigenvalues of these matrices, you need to find an eigenvector.
(b)
Hopefully you noticed that these matrices have the same eigenvalues but different eigenvectors. Let \(\vec{v}_1\) be the eigenvector of \(A\) corresponding to \(\lambda_1=\frac{1}{2}\) and \(\vec{v}_2\) be the eigenvector of \(A\) corresponding to \(\lambda_2=2\text{.}\) Let \(\vec{w}_1\) be the eigenvector of \(B\) corresponding to \(\lambda_1=\frac{1}{2}\) and \(\vec{w}_2\) be the eigenvector of \(B\) corresponding to \(\lambda_2=2\text{.}\)
Find a matrix \(C\) such that \(C \vec{v}_1 =\vec{w}_1\) and \(C \vec{v}_2=\vec{w}_2\text{.}\)
(c)
Find a matrix \(D\) such that \(D \vec{w}_1 =\vec{v}_1\) and \(D \vec{w}_2=\vec{v}_2\text{.}\)
(d)
Compute the matrix \(C A D\text{.}\)
(e)
Explain what just happened with matrix product \(C A D\text{...}\)
Activity3.2.2.
(a)
Let’s try to reverse engineer what just happened in the previous activity. Can we come up with the matrix that has eigenvalues \(\lambda_1=1.5\) and \(\lambda_2=-1/3\) with eigenvectors of \(\colvec{2 \\ 1} \) and \(\colvec{4 \\ 1}\text{?}\)
Think about what parts of the corresponding matrix parts are given by the information above. Once you have guessed at how to write the corresponding \(C A D\) matrix. Test your answer by checking what the eigenvalues and eigenvectors are.
