Choked flow

2014-01-20T17:09:35Z

86.183.12.111: /* Minimum pressure ratio required for choked flow to occur */ one criteria is a criterion

{{for|the theorem in algebraic number theory|Bauer's theorem}}
In [[mathematics]], the '''Bauer–Fike theorem''' is a standard result in the [[perturbation theory]] of the [[eigenvalue]] of a complex-valued [[diagonalizable|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 <math>A\in\mathbb{C}^{n,n}</math> be a [[diagonalizable|diagonalizable matrix]], and <math>V\in\mathbb{C}^{n,n}</math> be the non-singular [[eigenvector]] matrix such that <math>A=V\Lambda V^{-1}</math>. Moreover, let <math>\mu</math> be an eigenvalue of the matrix <math>A+\delta A</math>; then an eigenvalue <math>\lambda\in\sigma(A)</math> exists such that:

:<math>|\lambda-\mu|\leq\kappa_p (V)\|\delta A\|_p</math>

where <math>\kappa_p(V)=\|V\|_p\|V^{-1}\|_p</math> is the usual [[condition number]] in [[matrix norm|p-norm]].

===Proof===

If <math>\mu\in\sigma(A)</math>, we can choose <math>\lambda=\mu</math> and the thesis is trivially verified (since <math>\kappa_p(V)\geq 1</math>).

So, be <math>\mu\notin\sigma(A)</math>. Then <math>\det(\Lambda-\mu I)\ \ne\ 0</math>. <math>\mu</math> being an eigenvalue of <math>A+\delta A</math>, we have <math>\det(A+\delta A-\mu I)=0</math> and so

:<math>0=\det(V^{-1})\det(A+\delta A-\mu I)\det(V)=\det(\Lambda+V^{-1}\delta AV-\mu I)</math>
:<math>=\det(\Lambda-\mu I)\det[(\Lambda-\mu I)^{-1}V^{-1}\delta AV +I]</math>

and, since <math>\det(\Lambda-\mu I)\ \ne\ 0</math> as stated above, we must have

:<math>\det[(\Lambda-\mu I)^{-1}V^{-1}\delta AV +I]=\ 0</math>

which reveals the value −1 to be an eigenvalue of the matrix <math>(\Lambda-\mu I)^{-1}V^{-1}\delta AV</math>.

For each [[matrix norm|consistent matrix norm]], we have <math>|\lambda|\leq\|A\|</math>, so, all ''p''-norms being consistent, we can write:

:<math>1\leq\|(\Lambda-\mu I)^{-1}V^{-1}\delta AV\|_p\leq\|(\Lambda-\mu I)^{-1}\|_p\|V^{-1}\|_p\|V\|_p\|\delta A\|_p</math>
:<math>=\|(\Lambda-\mu I)^{-1}\|_p\ \kappa_p(V)\|\delta A\|_p</math>

But <math>(\Lambda-\mu I)^{-1}</math> being a diagonal matrix, the ''p''-norm is easily computed, and yields:

:<math>\|(\Lambda-\mu I)^{-1}\|_p\ =\max_{\|\mathbf{x}\|_p\ne 0} \frac{\|(\Lambda-\mu I)^{-1}\mathbf{x}\|_p}{\|\mathbf{x}\|_p}\ </math>
:<math>=\max_{\lambda\in\sigma(A)}\frac{1}{|\lambda -\mu|}\ =\ \frac{1}{\min_{\lambda\in\sigma(A)}|\lambda-\mu|}</math>

whence:

:<math>\min_{\lambda\in\sigma(A)}|\lambda-\mu|\leq\ \kappa_p(V)\|\delta A\|_p.\,</math>

The theorem can also be reformulated to better suit numerical methods.
In fact, dealing with real eigensystem problems, one often has an exact matrix <math>A</math>, but knows only an approximate eigenvalue-eigenvector couple, (<math>\tilde{\lambda}</math>,<math>\tilde{\mathbf{v}}</math>), and needs to bound the error. The following version comes in help.

==Theorem ([[Friedrich L. Bauer]], C.T.Fike – 1960) (alternative statement)==

Let <math>A\in\mathbb{C}^{n,n}</math> be a [[diagonalizable|diagonalizable matrix]], and be <math>V\in\mathbb{C}^{n,n}</math> the non singular [[eigenvector]] matrix such as <math>A=V\Lambda V^{-1}</math>. Be moreover (<math>\tilde{\lambda}</math>,<math>\mathbf{\tilde{v}}</math>) an approximate eigenvalue-eigenvector couple, and <math>\mathbf{r}=A\mathbf{\tilde{v}}-\tilde{\lambda}\mathbf{\tilde{v}}</math>; then an eigenvalue <math>\lambda\in\sigma(A)</math> exists such that:

:<math>|\lambda-\tilde{\lambda}|\leq\kappa_p (V)\frac{\|\mathbf{r}\|_p}{\|\mathbf{\tilde{v}}\|_p}</math>

where <math>\kappa_p(V)=\|V\|_p\|V^{-1}\|_p</math> is the usual [[condition number]] in [[matrix norm|p-norm]].

===Proof===

We solve this problem with Tarık's method:
m<math>\tilde{\lambda}\notin\sigma(A)</math> (otherwise, we can choose <math>\lambda=\tilde{\lambda}</math> and theorem is proven, since <math>\kappa_p(V)\geq 1</math>).
Then <math>(A-\tilde{\lambda} I)^{-1}</math> exists, so we can write:

:<math>\mathbf{\tilde{v}}=(A-\tilde{\lambda} I)^{-1}\mathbf{r}=V(D-\tilde{\lambda} I)^{-1}V^{-1}\mathbf{r}</math>

since <math>A</math> is diagonalizable; taking the p-norm of both sides, we obtain:

:<math>\|\mathbf{\tilde{v}}\|_p=\|V(D-\tilde{\lambda} I)^{-1}V^{-1}\mathbf{r}\|_p \leq \|V\|_p \|(D-\tilde{\lambda} I)^{-1}\|_p \|V^{-1}\|_p \|\mathbf{r}\|_p</math>
<math>=\kappa_p(V)\|(D-\tilde{\lambda} I)^{-1}\|_p \|\mathbf{r}\|_p.
</math>

But, since <math>(D-\tilde{\lambda} I)^{-1}</math> is a diagonal matrix, the p-norm is easily computed, and yields:

:<math>\|(D-\tilde{\lambda} I)^{-1}\|_p=\max_{\|\mathbf{x}\|_p \ne 0}\frac{\|(D-\tilde{\lambda} I)^{-1}\mathbf{x}\|_p}{\|\mathbf{x}\|_p}</math>
:<math>=\max_{\lambda\in\sigma(A)} \frac{1}{|\lambda-\tilde{\lambda}|}=\frac{1}{\min_{\lambda\in\sigma(A)}|\lambda-\tilde{\lambda}|}</math>

whence:

:<math>\min_{\lambda\in\sigma(A)}|\lambda-\tilde{\lambda}|\leq\kappa_p(V)\frac{\|\mathbf{r}\|_p}{\|\mathbf{\tilde{v}}\|_p}.</math>

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 <math>A\in\mathbb{C}^{n,n}</math> a non-singular, [[diagonalizable|diagonalizable matrix]], and be <math>V\in\mathbb{C}^{n,n}</math> the non singular [[eigenvector]] matrix such as <math>A=V\Lambda V^{-1}</math>. Be moreover <math>\mu</math> an eigenvalue of the matrix <math>A+\delta A</math>; then an eigenvalue <math>\lambda\in\sigma(A)</math> exists such that:

:<math>\frac{|\lambda-\mu|}{|\lambda|}\leq\kappa_p (V)\|A^{-1}\delta A\|_p</math>

(Note: <math>\|A^{-1}\delta A\|</math>can be formally viewed as the "relative variation of A", just as <math>|\lambda-\mu||\lambda|^{-1}</math> is the relative variation of λ.)

=== Proof ===
Since μ is an eigenvalue of (A+δA) and <math>det(A)\ne 0</math>, we have, left-multiplying by <math>-A^{-1}</math>:

:<math>-A^{-1}(A+\delta A)\mathbf{v}=-\mu A^{-1}\mathbf{v}</math>

that is, putting<math>\tilde{A}=\mu A^{-1}</math> and <math>\tilde{\delta A}=-A^{-1}\delta A</math>:

:<math>(\tilde{A}+\tilde{\delta A}-I)\mathbf{v}=\mathbf{0}</math>

which means that<math>\tilde{\mu}=1</math>is an eigenvalue of<math>(\tilde{A}+\tilde{\delta A})</math>, with <math>\mathbf{v}</math>eigenvector. Now, the eigenvalues of <math>\tilde{A}</math>are <math>\frac{\mu}{\lambda_i}</math>, while its eigenvector matrix is the same as A. Applying the Bauer–Fike theorem to the matrix<math>\tilde{A}+\tilde{\delta A}</math> and to its eigenvalue<math>\tilde{\mu}=1</math>, we obtain:

:<math>\min_{\lambda\in\sigma(A)}\left|\frac{\mu}{\lambda}-1\right|=\min_{\lambda\in\sigma(A)}\frac{|\lambda-\mu|}{|\lambda|}\leq\kappa_p (V)\|A^{-1}\delta A\|_p</math>

== Remark ==

If A is [[normal matrix|normal]], V is a [[unitary matrix]], and <math>\|V\|_2=\|V^{-1}\|_2=1</math>, so that <math>\kappa_2(V)=1</math>.

The Bauer–Fike theorem then becomes:

:<math>\exists\lambda\in\sigma(A): |\lambda-\mu|\leq\|\delta A\|_2</math>

:( <math>\exists\lambda\in\sigma(A): |\lambda-\tilde{\lambda}|\leq\frac{\|\mathbf{r}\|_2}{\|\mathbf{\tilde{v}}\|_2}</math> 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's inequality|Weyl's theorem on eigenvalues]].

== 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

{{DEFAULTSORT:Bauer-Fike theorem}}
[[Category:Spectral theory]]
[[Category:Theorems in analysis]]
[[Category:Articles containing proofs]]

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