Question

Exercises \({\bf{13}}\) and \(14\) apply to a \({\bf{3 \times 3}}\) matrix \(A\) whose eigenvalues are estimated to be \({\bf{4}}\), \({\bf{4}}\), and \({\bf{3}}\).

14. Suppose the eigenvalues close to \({\bf{4}}\) and \({\bf{ - 4}}\) are known to have exactly the same absolute value. Describe how one might obtain a sequence that estimates the eigenvalue close to \({\bf{4}}\).

Short answer

The power method will not work however, the inverse power method can be used. If the initial estimate is chosen near the eigenvalue close to \(4\), then the inverse power method should produce a sequence that estimates the eigenvalue close to \(4\).

Step-by-step solution

Write the definition of the Power Method

The Power Method: An \(n \times n\) matrix \(A\) with a strictly dominant eigenvalue \({\lambda _1}\) that means \({\lambda _1}\) must be larger in absolute value than all the other eigenvalues.

Check whether the power method will work or not

If the eigenvalues are close to\(4\)and\( - 4\)have the same absolute values, then none of these is a strictly dominant eigenvalue, so the power method will not work.

However, the inverse power method can be used. If the initial estimate is chosen near the eigenvalue close to \(4\), then the inverse power method should produce a sequence that estimates the eigenvalue close to \(4\).