Activity 5.2.4
Remember that the absolute value of a number tells us how far that number is from \(0\) on the real number line. We may therefore think of the inverse power method as telling us the eigenvalue closest to \(0\text{.}\)
If \(\vvec\) is an eigenvalue of \(A\) with associated eigenvalue \(\lambda\text{,}\) explain why \(\vvec\) is an eigenvector of \(A  sI\) where \(s\) is some scalar.
What is the eigenvalue of \(AsI\) associated to the eigenvector \(\vvec\text{?}\)
Explain why the eigenvalue of \(A\) closest to \(s\) is the eigenvalue of \(AsI\) closest to \(0\text{.}\)
Explain why applying the inverse power method to \(AsI\) gives the eigenvalue of \(A\) closest to \(s\text{.}\)

Consider the matrix \(A = \left[\begin{array}{rrrr} 3.6 \amp 1.6 \amp 4.0 \amp 7.6 \\ 1.6 \amp 2.2 \amp 4.4 \amp 4.1 \\ 3.9 \amp 4.3 \amp 9.0 \amp 0.6 \\ 7.6 \amp 4.1 \amp 0.6 \amp 5.0 \\ \end{array}\right] \text{.}\) If we use the power method and inverse power method, we find two eigenvalues, \(\lambda_1=16.35\) and \(\lambda_2=0.75\text{.}\) Viewing these eigenvalues on a number line, we know that the other eigenvalues lie in the range between \(\lambda_1\) and \(\lambda_1\text{,}\) as shaded in FigureĀ 5.2.1.
The Sage cell below has a function
find_closest_eigenvalue(A, s, x, N)
that implements \(N\) steps of the inverse power method using the matrix \(AsI\) and an initial vector \(x\text{.}\) This function prints approximations to the eigenvalues and eigenvectors of \(A\text{.}\) By trying different values in the gray regions of the number line, find the other two eigenvalues of \(A\text{.}\) Write a list of the four eigenvalues of \(A\) in increasing order.