The Power Method is an iterative algorithm used to find the dominant eigenvalue (the largest in magnitude) and its corresponding eigenvector for a given square matrix.

Steps to Apply the Power Method:

1. Choose an initial guess vector (usually a random vector or a vector with all components equal to 1).
2. Multiply the matrix $A$ by the current vector.
3. Normalize the resulting vector to avoid numerical instability.
4. Repeat the process until convergence (i.e., when the vector changes very little between iterations).
5. Approximate the dominant eigenvalue as the Rayleigh quotient $\lambda_k = \frac{\mathbf{x}_k^T A \mathbf{x}_k}{\mathbf{x}_k^T \mathbf{x}_k}$, where $\mathbf{x}_k$ is the vector after iteration $k$.

Now, let's solve for the dominant eigenvalue and eigenvector using the matrix $A = \begin{bmatrix} 1 & 3 & -1 \\ 3 & 2 & 4 \\ -1 & 4 & 10 \end{bmatrix}$.

I will perform these calculations iteratively using Python.The dominant eigenvalue of the matrix $A = \begin{bmatrix} 1 & 3 & -1 \\ 3 & 2 & 4 \\ -1 & 4 & 10 \end{bmatrix}$ is approximately 11.662.

The corresponding eigenvector is approximately: $\mathbf{v} = \begin{bmatrix} 0.0229 \\ 0.3885 \\ 0.9212 \end{bmatrix}$

This result was obtained after 12 iterations of the Power Method.

Would you like more details on any part of the process, or have any questions?

Here are five related questions you may find interesting:

1. How does the Power Method work for matrices with complex eigenvalues?
2. Can the Power Method be used to find the smallest eigenvalue of a matrix?
3. What are the advantages of using the Power Method over other methods for large matrices?
4. How would the Power Method perform if the matrix $A$ had repeated eigenvalues?
5. How can the convergence rate of the Power Method be improved?

Tip: The convergence rate of the Power Method depends on the ratio of the magnitudes of the dominant and the second-largest eigenvalues. The closer they are, the slower the convergence.