Bauer–Fike theorem
In mathematics, the Bauer–Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact matrix. Informally speaking, what it says is that the sensitivity of the eigenvalues is estimated by the condition number of the matrix of eigenvectors. The theorem was proved by Friedrich L. Bauer and C. T. Fike in 1960. The setup
In what follows we assume that:
  is a diagonalizable matrix;
  is the non-singular eigenvector matrix such that , where  is a diagonal matrix. If  is invertible, its condition number in -norm is denoted by  and defined by:

The Bauer–Fike Theorem
Bauer–Fike Theorem. Let  be an eigenvalue of . Then there exists  such that:

Proof. We can suppose , otherwise take  and the result is trivially true since . Since  is an eigenvalue of , we have  and so

However our assumption, , implies that:  and therefore we can write:

This reveals  to be an eigenvalue of 

Since all -norms are consistent matrix norms we have  where  is an eigenvalue of . In this instance this gives us:

But  is a diagonal matrix, the -norm of which is easily computed:

whence:

An Alternate Formulation
The theorem can also be reformulated to better suit numerical methods. In fact, dealing with real eigensystem problems, one often has an exact matrix , but knows only an approximate eigenvalue-eigenvector couple,  and needs to bound the error. The following version comes in help. Bauer–Fike Theorem (Alternate Formulation). Let  be an approximate eigenvalue-eigenvector couple, and . Then there exists  such that:

Proof. We can suppose , otherwise take  and the result is trivially true since . So  exists, so we can write:

since  is diagonalizable; taking the -norm of both sides, we obtain:

However 

 

is a diagonal matrix and its -norm is easily computed:

whence:

A Relative Bound
Both formulations of Bauer–Fike theorem yield an absolute bound. The following corollary is useful whenever a relative bound is needed:

Corollary. Suppose  is invertible and that  is an eigenvalue of . Then there exists  such that:

Note. can be formally viewed as the relative variation of , just as  is the relative variation of . Proof. Since  is an eigenvalue of  and , by multiplying by  from left we have:

If we set:

then we have:

which means that  is an eigenvalue of , with  as an eigenvector. Now, the eigenvalues of  are , while it has the same eigenvector matrix as . Applying the Bauer–Fike theorem to  with eigenvalue , gives us:

The Case of Normal Matrices 
If  is normal,  is a unitary matrix, therefore:

so that . The Bauer–Fike theorem then becomes:

Or in alternate formulation:

which obviously remains true if  is a Hermitian matrix. In this case, however, a much stronger result holds, known as the Weyl's theorem on eigenvalues. In the hermitian case one can also restate the Bauer–Fike theorem in the form that the map  that maps a matrix to its spectrum is a non-expansive function with respect to the Hausdorff distance on the set of compact subsets of . References 
 
 

Spectral theory
Theorems in analysis
Articles containing proofs