Math Problem Statement

Using power method, find the dominant eigen value and the corresponding eigen vector of A = [[1, 3, - 1], [3, 2, 4], [- 1, 4, 10]]

Solution

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

The steps for the Power Method are as follows:

  1. Initialize a vector x0x_0 (a random non-zero vector).
  2. Multiply the matrix AA by the vector xkx_k at each iteration to obtain a new vector xk+1=Axkx_{k+1} = A x_k.
  3. Normalize the resulting vector xk+1x_{k+1}.
  4. Repeat the process until the difference between successive iterations becomes very small (convergence).
  5. Estimate the dominant eigenvalue as the ratio between successive vectors.

Given matrix AA:

1 & 3 & -1 \\ 3 & 2 & 4 \\ -1 & 4 & 10 \end{bmatrix}$$ Let's go step by step and calculate the dominant eigenvalue and the corresponding eigenvector. ### 1. Initial Guess We begin with an initial guess for the eigenvector, say: $$x_0 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$$ ### 2. Iterative Steps Now, we will iteratively apply the matrix $$A$$ to this vector, normalizing it at each step. I'll compute the first few iterations and update the dominant eigenvalue and eigenvector using the power method. Let me proceed with the calculations.The dominant eigenvalue of the matrix $$A$$ is approximately $$11.662$$, and the corresponding normalized eigenvector is: $$v = \begin{bmatrix} 0.0229 \\ 0.3885 \\ 0.9212 \end{bmatrix}$$ This eigenvector corresponds to the dominant eigenvalue $$11.662$$. Would you like further details on the calculation, or do you have any other questions? Here are five related questions for further exploration: 1. How does the Power Method work for non-symmetric matrices? 2. What are the limitations of the Power Method in finding eigenvalues? 3. Can you explain how the Rayleigh quotient is used to estimate eigenvalues? 4. How does matrix normalization affect the convergence speed of the Power Method? 5. Can the Power Method be applied to find the smallest eigenvalue of a matrix? **Tip**: The Power Method converges fastest when there is a large gap between the dominant eigenvalue and the other eigenvalues.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Eigenvalues
Eigenvectors
Iterative Methods

Formulas

Power Method Algorithm: x_{k+1} = A * x_k, λ ≈ (x_k+1^T * A * x_k) / (x_k+1^T * x_k)

Theorems

Power Method for finding dominant eigenvalue and eigenvector

Suitable Grade Level

Undergraduate (Linear Algebra)