Find the dominant eigen value of A = [[1, 6, 1], [1, 2, 0], [0, 0, 3]] using power method 

The **power method** is an iterative technique used to approximate the dominant (largest) eigenvalue of a matrix. Let's outline the steps and compute the dominant eigenvalue for the given matrix:

### Matrix:
\[
A = \begin{bmatrix} 1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}
\]

### Step-by-Step Solution:
1. **Choose an initial vector** \( x^{(0)} \).
   - Typically, we choose \( x^{(0)} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \).

2. **Normalize the initial vector** \( x^{(0)} \).
   - Compute the norm of \( x^{(0)} \):  
     \[
     \| x^{(0)} \| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}
     \]
   - Normalize:  
     \[
     x^{(0)} = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}
     \]

3. **Iteratively compute the matrix-vector product** \( y^{(k)} = A x^{(k)} \) until convergence.
   - At each step, update the vector:  
     \[
     x^{(k+1)} = \frac{y^{(k)}}{\| y^{(k)} \|}
     \]
   - Approximate the dominant eigenvalue as:  
     \[
     \lambda^{(k)} = \frac{y^{(k)} \cdot x^{(k)}}{x^{(k)} \cdot x^{(k)}}
     \]

4. **Continue iterating until** \( x^{(k)} \) **converges to a fixed vector, or until** \( \lambda^{(k)} \) **stabilizes**.

### Let's solve this problem using these steps.

I'll implement the power method to find the dominant eigenvalue for matrix \( A \).

Find the dominant eigenvalue of A = [[1, 6, 1], [1, 2, 0], [0, 0, 3]] using power method

Learn how to find the dominant eigenvalue of the matrix A = [[1, 6, 1], [1, 2, 0], [0, 0, 3]] using the power method. This step-by-step solution explains key concepts in linear algebra, including eigenvalues, eigenvectors, and iterative methods. Ideal for undergraduate students.

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