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To find the dominant eigenvalue and the corresponding eigenvector of the matrix \( A \) using the power method, we can follow these steps iteratively. ### Given Matrix: \[ A = \begin{bmatrix} -4 & 14 & 0 \\ -5 & 13 & 0 \\ -1 & 0 & 2 \end{bmatrix} \] ### Initial Vector: \[ y^0 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \] ### Steps for Power Method: 1. Initialize the vector \( y^0 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \). 2. Iterate for 5 steps: - Multiply the matrix \( A \) by the vector \( y^{k-1} \) to get a new vector \( y^k \). - Normalize the new vector to get the next vector. - The dominant eigenvalue approximation \( \lambda^k \) is calculated by taking the ratio of the new and old vectors' components. ### Iteration Details: 1. First Iteration: \[ y^1 = A y^0 = \begin{bmatrix} -4 & 14 & 0 \\ -5 & 13 & 0 \\ -1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ ________ _________ __________ ________ ____ ______ ___ __________ ______ ____ _________.
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