To find the dominant eigenvalue and its corresponding eigenvector of the matrix $A = \begin{bmatrix} 1 & 3 & -1 \\ 3 & 2 & 4 \\ -1 & 4 & 10 \end{bmatrix}$ using the Power Method, we follow an iterative process that starts with an arbitrary non-zero vector and repeatedly multiplies it by the matrix. The sequence of vectors will converge to the eigenvector corresponding to the dominant eigenvalue (the largest eigenvalue in magnitude).

Steps of the Power Method:

1. Start with an initial guess for the eigenvector $x_0$. Let’s pick $x_0 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$, but any non-zero vector can work.
2. Normalize the vector after each multiplication.
3. Repeat the process until convergence, or until the eigenvalue (the ratio of elements of successive vectors) and the eigenvector stabilize.

Step-by-Step Computation

1. Initial guess: $x_0 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$

2. First iteration: Multiply the matrix $A$ by the vector $x_0$: $x_1 = A \cdot x_0 = \begin{bmatrix} 1 & 3 & -1 \\ 3 & 2 & 4 \\ -1 & 4 & 10 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 9 \\ 13 \end{bmatrix}$ Normalize $x_1$: $\|x_1\| = \sqrt{3^2 + 9^2 + 13^2} = \sqrt{9 + 81 + 169} = \sqrt{259}$ Normalized vector $x_1$ is: $x_1 = \frac{1}{\sqrt{259}} \begin{bmatrix} 3 \\ 9 \\ 13 \end{bmatrix}$

3. Second iteration: Multiply $A$ by $x_1$: $x_2 = A \cdot x_1 = \begin{bmatrix} 1 & 3 & -1 \\ 3 & 2 & 4 \\ -1 & 4 & 10 \end{bmatrix} \cdot \begin{bmatrix} 3 \\ 9 \\ 13 \end{bmatrix} = \begin{bmatrix} 12 \\ 59 \\ 107 \end{bmatrix}$ Normalize $x_2$: $\|x_2\| = \sqrt{12^2 + 59^2 + 107^2} = \sqrt{144 + 3481 + 11449} = \sqrt{15074}$ Normalized vector $x_2$ is: $x_2 = \frac{1}{\sqrt{15074}} \begin{bmatrix} 12 \\ 59 \\ 107 \end{bmatrix}$

4. Continue the iterations: Repeat this process until the vectors stabilize, i.e., the ratio of successive eigenvalue approximations converges.

Each multiplication brings the result closer to the dominant eigenvector, and the ratio between corresponding elements of successive vectors gives an approximation of the dominant eigenvalue.

Would you like to proceed with more detailed iterations or further clarifications?

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Here are five questions to deepen understanding:

1. Why is the Power Method best for finding the dominant eigenvalue?
2. How does the choice of the initial vector impact the speed of convergence in the Power Method?
3. Can the Power Method detect non-dominant eigenvalues?
4. How does normalization affect the iterative process of the Power Method?
5. What are the conditions for the Power Method to converge successfully?

Tip: Ensure the matrix is diagonally dominant or the dominant eigenvalue has a large magnitude difference from the others for faster convergence in the Power Method.