Symmetric Eigenvalue Deflation

Symmetric eigenvalue deflation is a technique used in numerical linear algebra to find multiple eigenvalues of a symmetric matrix. Once an eigenvalue and its corresponding eigenvector are found, the matrix can be modified (deflated) so that the found eigenvalue is shifted, while the other eigenvalues and eigenvectors remain unchanged.

Worked Example

Consider a matrix A with eigenvalues 5, 2, and -1. The dominant eigenvalue is 5, with corresponding unit eigenvector q_1. We can deflate the matrix by constructing:

D = A - α q_1 q_1^T

If we choose α = 5, the new eigenvalues of D will be 5 - 5 = 0, 2, and -1. The dominant magnitude eigenvalue of D is now 2. This isolates the remaining spectrum without changing the other eigenpairs.

Experiment

Use the Symmetric Eigenvalue Deflation interactive module to explore this concept.

Select 1 for the Subtraction factor.
Observe that the Removed eigenvalue amount is 5.
Verify that the Dominant remaining eigenvalue is exactly 2.
Select 1.5 for the Subtraction factor.
Observe that the Removed eigenvalue amount becomes 7.5.
Notice that the Dominant remaining eigenvalue is now -2.5, which has a magnitude of 2.5.

Limitations

While exact deflation works perfectly in theory, in practice with floating-point arithmetic, small errors in the computed eigenvector q_1 can introduce perturbations in the remaining eigenvalues. Furthermore, overshooting the subtraction factor can make the deflated eigenvalue re-emerge as the dominant magnitude eigenvalue with a negative sign, making it harder to find the true next eigenvalue.

References

For more details on eigenvalue methods and deflation, consult the following resources.

Power Method and Deflation (Computational Physics)

Symmetric Eigenvalue Deflation