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".

Theorem (Friedrich L. Bauer, C.T.Fike - 1960)

Let $Ainmathbb\{C\}^\{n,n\}$ be a diagonalizable matrix, and be $Vinmathbb\{C\}^\{n,n\}$ the non singular eigenvector matrix such that $A=VLambda\; V^\{-1\}$. Be moreover $mu$ an eigenvalue of the matrix $A+delta\; A$; then an eigenvalue $lambdainsigma(A)$ exists such that:

:$|lambda-mu|leqkappa\_p\; (V)|delta\; A|\_p$

where $kappa\_p(V)=|V|\_p|V^\{-1\}|\_p$ is the usual condition number in p-norm.

Proof

If $muinsigma(A)$, we can choose $lambda=mu$ and the thesis is trivially verified (since $kappa\_p(V)geq\; 1$).

So, be $mu\; otinsigma(A)$. Then $det(Lambda-mu\; I)\; e\; 0$. $mu$ being an eigenvalue of $A+delta\; A$, we have $det(A+delta\; A-mu\; I)=0$ and so

:$0=\; det(V^\{-1\})\; det(A+delta\; A-mu\; I)\; det(V)=det(Lambda+V^\{-1\}delta\; AV-mu\; I)=$:$=det(Lambda-mu\; I)\; det\; [(Lambda-mu\; I)^\{-1\}V^\{-1\}delta\; AV\; +I]$

and, since $det(Lambda-mu\; I)\; e\; 0$ as stated above, we must have

:$det\; [(Lambda-mu\; I)^\{-1\}V^\{-1\}delta\; AV\; +I]\; =\; 0$

which reveals the value -1 to be an eigenvalue of the matrix $(Lambda-mu\; I)^\{-1\}V^\{-1\}delta\; AV$.

For each consistent matrix norm, we have $|lambda|leq|A|$, so, being all p-norms consistent, we can write:

:$1leq|(Lambda-mu\; I)^\{-1\}V^\{-1\}delta\; AV|\_pleq|(Lambda-mu\; I)^\{-1\}|\_p|V^\{-1\}|\_p|V|\_p|delta\; A|\_p=$:$=|(Lambda-mu\; I)^\{-1\}|\_p\; kappa\_p(V)|delta\; A|\_p$

But $(Lambda-mu\; I)^\{-1\}$ being a diagonal matrix, the p-norm is easily computed, and yields:

:$|(Lambda-mu\; I)^\{-1\}|\_p\; =\; max\_\{|mathbf\{x\}|\_p\; e\; 0\}\; frac\{|(Lambda-mu\; I)^\{-1\}mathbf\{x\}|\_p\}\{|mathbf\{x\}|\_p\}\; =$:$=max\_\{lambdainsigma(A)\}frac\{1\}$

whence:

:$min\_\{lambdainsigma(A)\}|lambda-\; ilde\{lambda\}|leqkappa\_p(V)frac\{|mathbf\{r\}|\_p\}\{|mathbf\{\; ilde\{v|\_p\}.$

The Bauer-Fike theorem, in both versions, yields an absolute bound. The following corollary, which, besides all the hypothesis of Bauer-Fike theorem, requires also the non-singularity of A, turns out to be useful whenever a relative bound is needed.

Corollary

Be $Ainmathbb\{C\}^\{n,n\}$ a non-singular, diagonalizable matrix, and be $Vinmathbb\{C\}^\{n,n\}$ the non singular eigenvector matrix such as $A=VLambda\; V^\{-1\}$. Be moreover $mu$ an eigenvalue of the matrix $A+delta\; A$; then an eigenvalue $lambdainsigma(A)$ exists such that:

:$fracleqkappa\_p\; (V)|A^\{-1\}delta\; A|\_p$

Remark

If A is normal, V is a unitary matrix, and $|V|\_2=|V^\{-1\}|\_2=1$, so that $kappa\_2(V)=1$.

The Bauer-Fike theorem then becomes:

:$existslambdainsigma(A):\; |lambda-mu|leq|delta\; A|\_2$

:($existslambdainsigma(A):\; |lambda-\; ilde\{lambda\}|leqfrac\{|mathbf\{r\}|\_2\}\{|mathbf\{\; ilde\{v|\_2\}$ in the alternative formulation)

which obviously remains true if A is a Hermitian matrix. In this case, however, a much stronger result holds, known as the Weyl theorem.

References

# F. L. Bauer and C. T. Fike. "Norms and exclusion theorems". Numer. Math. 2 (1960), 137-141.

# S. C. Eisenstat and I. C. F. Ipsen. "Three absolute perturbation bounds for matrix eigenvalues imply relative bounds". SIAM Journal on Matrix Analysis and Applications Vol. 20, N. 1 (1998), 149-158