Question

In the power method, we compute eigenvectors using a symmetric matrix Γ. Determine the steps for computing the eigenvector of a symmetric matrix Γ.

Answer 1

To compute the eigenvector of a symmetric matrix using the power method, begin with an initial vector, normalize it, iterate through matrix-vector multiplications and normalization, continue until convergence, and then estimate the eigenvalue.

Step-by-step explanation:

To compute the eigenvector of a symmetric matrix Γ using the power method, one should follow these general steps:

1. Choose an initial vector x₀ that is not orthogonal to the true eigenvector.
2. Normalize x₀ to have a norm (length) of 1. This is usually done using the Euclidean norm.
3. Iterate the process xₙ₊₁ = Γxₙ, then normalize xₙ₊₁ to a unit vector after each multiplication.
4. Convergence: Continue the iteration until the changes in the estimated eigenvectors between two successive iterations are below a certain tolerance level.
5. Estimate the eigenvalue: Once the corresponding eigenvector converges, the associated eigenvalue can be estimated by using the Rayleigh quotient xᵗᵒ⁻ Γxᵗ.

It is important to note that the power method, used as described, will find the eigenvalue and eigenvector of greatest magnitude, which are called the principal eigenvalue and eigenvector, respectively, when the matrix is symmetric.