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 \({\bf{3}}\).

13. If the eigenvalues close to \({\bf{4}}\) and \( - 4\) are known to have different absolute values, will the power method work? Is it likely to be useful?

Short answer

If the ratio \(\left| {\frac{{{\lambda _2}}}{{{\lambda _1}}}} \right|\) is close to \(1\) then the sequences \(\left\{ {{\mu _k}} \right\}\) and \(\left\{ {{x_k}} \right\}\) can converge very slowly and the method is not useful.

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 different absolute values, then one of these is a strictly dominant eigenvalue, so the power method will work.

If the ratio\(\left| {\frac{{{\lambda _2}}}{{{\lambda _1}}}} \right|\)is close to\(1\)then the sequences\(\left\{ {{\mu _k}} \right\}\)and\(\left\{ {{x_k}} \right\}\)can converge very slowly and the method is not useful.