Using power method, find the dominant eigen value and the corresponding eigen vector of

A = [[1, 3, - 1], [3, 2, 4], [- 1, 4, 10]]

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.

Using power method, find the dominant eigenvalue and the corresponding eigenvector of A = [[1, 3, -1], [3, 2, 4], [-1, 4, 10]].

This guide demonstrates how to find the dominant eigenvalue and the corresponding eigenvector of matrix A = [[1, 3, -1], [3, 2, 4], [-1, 4, 10]] using the Power Method. The problem covers key linear algebra concepts, including matrix operations and the Rayleigh Quotient formula. Suitable for undergraduate mathematics students.

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