Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds

We show that three well-known perturbation bounds for matrix eigenvalues imply relative bounds: the Bauer--Fike and Hoffman--Wielandt theorems for diagonalizable matrices, and Weyl's theorem for Hermitian matrices. As a consequence, relative perturbation bounds are not necessarily stronger than absolute bounds, and the conditioning of an eigenvalue in the relative sense is the same as in the absolute sense. We also show that eigenvalues of normal matrices are no more sensitive to perturbations than eigenvalues of Hermitian positive-definite matrices. The relative error bounds are invariant under congruence transformations, such as grading and scaling.

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