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Chapter 5: Q14E (page 267)

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

Expert verified

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

01

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.

02

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 \(4\).

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Most popular questions from this chapter

a. Let \(A\) be a diagonalizable \(n \times n\) matrix. Show that if the multiplicity of an eigenvalue \(\lambda \) is \(n\), then \(A = \lambda I\).

b. Use part (a) to show that the matrix \(A =\left({\begin{aligned}{*{20}{l}}3&1\\0&3\end{aligned}}\right)\) is not diagonalizable.

Compute the quantities in Exercises 1-8 using the vectors

\({\mathop{\rm u}\nolimits} = \left( {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right),{\rm{ }}{\mathop{\rm v}\nolimits} = \left( {\begin{array}{*{20}{c}}4\\6\end{array}} \right),{\rm{ }}{\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}3\\{ - 1}\\{ - 5}\end{array}} \right),{\rm{ }}{\mathop{\rm x}\nolimits} = \left( {\begin{array}{*{20}{c}}6\\{ - 2}\\3\end{array}} \right)\)

3. \(\frac{1}{{{\mathop{\rm w}\nolimits} \cdot {\mathop{\rm w}\nolimits} }}{\mathop{\rm w}\nolimits} \)

Question 18: It can be shown that the algebraic multiplicity of an eigenvalue \(\lambda \) is always greater than or equal to the dimension of the eigenspace corresponding to \(\lambda \). Find \(h\) in the matrix \(A\) below such that the eigenspace for \(\lambda = 5\) is two-dimensional:

\[A = \left[ {\begin{array}{*{20}{c}}5&{ - 2}&6&{ - 1}\\0&3&h&0\\0&0&5&4\\0&0&0&1\end{array}} \right]\]

Question: Find the characteristic polynomial and the eigenvalues of the matrices in Exercises 1-8.

4. \(\left[ {\begin{array}{*{20}{c}}5&-3\\-4&3\end{array}} \right]\)

Question: Construct a random integer-valued \(4 \times 4\) matrix \(A\), and verify that \(A\) and \({A^T}\) have the same characteristic polynomial (the same eigenvalues with the same multiplicities). Do \(A\) and \({A^T}\) have the same eigenvectors? Make the same analysis of a \(5 \times 5\) matrix. Report the matrices and your conclusions.

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