# Condition number of simple eigenvalues

• May 16th 2011, 09:25 AM
Mollier
Condition number of simple eigenvalues
Hi, there's a step in the derivation of the condition number that I do not understand.

Let $\displaystyle A\in\mathbb{C}^{n\times n}$ and let $\displaystyle \lambda$ be a simple eigenvalue of $\displaystyle A$.
Let $\displaystyle u$ be a right eigenvector and perturbe such that

$\displaystyle (A+\delta A)(u +\delta u) = (\lambda +\delta\lambda)(u + \delta u).$

I multiply this thing and use the fact that $\displaystyle Au=\lambda u$ to get,

$\displaystyle A(\delta u) +(\delta A)u = \lambda(\delta u) + (\delta\lambda)u.$

Now let $\displaystyle v^*$ be a left eigenvector of $\displaystyle A$ and multiply the above equation from the left. Using the fact that $\displaystyle v^*A=v^*\lambda$ we can write

$\displaystyle v^*(\delta A)u = v^*(\delta\lambda)u,$

hence,

$\displaystyle |\delta\lambda| = \frac{|v^*(\delta A)u|}{|v^*u|}.$

By assuming that $\displaystyle ||u||_2=||v||_2=1$ we can bound this as

$\displaystyle |\delta\lambda| \leq \frac{||\delta A||_2}{|v^* u|}.$

I hope you are still awake :) What happens next is confusing me. If $\displaystyle \lambda\neq 0$ then

$\displaystyle \frac{|\delta\lambda|}{|\lambda|}\leq \frac{||A||_2}{|\lambda||v^*u|}\cdot\frac{||\delta A||_2}{||A||_2},$

and therefore

$\displaystyle cond(\lambda) = \frac{||A||_2}{|\lambda| |v^*u|}.$

I do not understand the (and therefore) part :) Is this some alternative definition of condition number? Hope someone has the patience to help me out, thanks!
• May 29th 2011, 02:23 PM
ojones
This doesn't look quite right to me either. The condition number of a simple eigenvalue is $\displaystyle 1/|v^*u|$. What's the reference here?
Now that I think about it, the quantities given are relative errors. This might be a kind of relative condition number, if such a notion exists.