Condition number of simple eigenvalues
Hi, there's a step in the derivation of the condition number that I do not understand.
Let and let be a simple eigenvalue of .
Let be a right eigenvector and perturbe such that
I multiply this thing and use the fact that to get,
Now let be a left eigenvector of and multiply the above equation from the left. Using the fact that we can write
By assuming that we can bound this as
I hope you are still awake :) What happens next is confusing me. If then
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!