Thread: Condition number of simple eigenvalues

1. Condition number of simple eigenvalues

Hi, there's a step in the derivation of the condition number that I do not understand.

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

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

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

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

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

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

hence,

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

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

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

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

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

and therefore

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

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!

2. This doesn't look quite right to me either. The condition number of a simple eigenvalue is $1/|v^*u|$. 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.