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Great! We are asked to find the dominant eigenvalue and eigenvector of $$ A = \begin{bmatrix}-4 & 14 & 0 \\ -5 & 13 & 0 \\ -1 & 0 & 2\end{bmatrix} $$ using five iterations of the power method, starting with $$ y^{(0)} = \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}. $$ Let's solve step by step manually. --- ### Step 1: Power Method Overview The power method iteratively multiplies the matrix $A$ with the vector $y^{(k)}$: $$ y^{(k+1)} = A y^{(k)} $$ After each iteration, we normalize $y^{(k+1)}$ by dividing by the component with the largest absolute value. The ratio of the largest component of $y^{(k+1)}$ to the corresponding component of $y^{(k)}$ _____ ___ _________ ______ __________ ________ ________ __________.
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