##### Activity5.2.2

Let's begin by considering the matrix $$A = \left[\begin{array}{rr} 0.5 \amp 0.2 \\ 0.4 \amp 0.7 \\ \end{array}\right]$$ and the initial vector $$\xvec_0 = \twovec{1}{0}\text{.}$$

1. Compute the vector $$\xvec_1 = A\xvec_0\text{.}$$

2. Find $$m_1\text{,}$$ the component of $$\xvec_1$$ that has the largest absolute value. Then form $$\overline{\xvec}_1 = \frac 1{m_1} \xvec_1\text{.}$$ Notice that the component having the largest absolute value of $$\overline{\xvec}_1$$ is $$1\text{.}$$

3. Find the vector $$\xvec_2 = A\overline{\xvec}_1\text{.}$$ Identify the component $$m_2$$ of $$\xvec_2$$ having the largest absolute value. Then form $$\overline{\xvec}_2 = \frac1{m_2}\overline{\xvec}_1$$ to obtain a vector in which the component with the largest absolute value is $$1\text{.}$$

4. The Sage cell below defines a function that implements the power method. Define the matrix $$A$$ and initial vector $$\xvec_0$$ below. The command power(A, x0, N) will print out the multiplier $$m$$ and the vectors $$\overline{\xvec}_k$$ for $$N$$ steps of the power method.

How does this computation identify an eigenvector of the matrix $$A\text{?}$$

5. What is the corresponding eigenvalue of this eigenvector?

6. How do the values of the multipliers $$m_k$$ tell us the eigenvalue associated to the eigenvector we have found?

7. Consider now the matrix $$A=\left[\begin{array}{rr} -5.1 \amp 5.7 \\ -3.8 \amp 4.4 \\ \end{array}\right] \text{.}$$ Use the power method to find the dominant eigenvalue of $$A$$ and an associated eigenvector.

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