Activity 5.2.2
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 \(\xvec_0 = \twovec{1}{0}\text{.}\)
Compute the vector \(\xvec_1 = A\xvec_0\text{.}\)
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 \(1\text{.}\)
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 \(1\text{.}\)

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 \(A\text{?}\)
What is the corresponding eigenvalue of this eigenvector?
How do the values of the multipliers \(m_k\) tell us the eigenvalue associated to the eigenvector we have found?
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.