Bauer–Fike Nonnormal Bound

The Bauer-Fike theorem provides bounds on the eigenvalues of perturbed matrices. In the case of nonnormal matrices, the sensitivity of eigenvalues to perturbations is amplified by the condition number of the eigenvector matrix.

Theory

Let A be a diagonalizable matrix such that A = VDV^(-1). For any perturbation E, the eigenvalues of A + E are bounded. According to the Bauer-Fike theorem, each perturbed eigenvalue lies in the union of disks centered at the original eigenvalues of A, with radius equal to the condition number of V multiplied by the norm of E.

Worked Example

Consider the matrix A and its perturbation. We can calculate the bounds explicitly to see the effects.

Original eigenvalues: 1 and 3
Perturbation norm: 0.25
Condition number of V: 2.618033988749895
Calculated Bauer-Fike Radius: 0.6545084971874737

Experiment

Adjust the parameters to see the effects of nonnormality in the module:

Change the Eigenbasis shear to increase the nonnormality of the matrix.
Modify the Perturbation delta to alter the perturbation matrix E.
Observe how the condition number of V inflates the Bauer-Fike radius.
Notice when the eigenvalues become complex or defective (non-diagonalizable).

Limitations

The Bauer-Fike bound is often pessimistic for highly nonnormal matrices. Pseudospectra can provide a more accurate picture of the eigenvalue sensitivity in such cases. The bound also requires the matrix A to be diagonalizable.

References

The Bauer-Fike theorem is a fundamental result in numerical linear algebra, providing a theoretical foundation for understanding eigenvalue sensitivity. This module explores its implications for nonnormal matrices, where the condition number of the eigenbasis can significantly amplify the effects of perturbations. For further reading and a deeper mathematical treatment, see the following resource:Eigenvalues and the Bauer-Fike Theorem.

Bauer–Fike Nonnormal Bound