Calculating the Inverse of Operator L in R^2

[Bimmel]

hello,

given is the Operator L=$$\widehat{1}$$+|u><v|, where $$\widehat{1}$$ means the unity-tensor.

Whats the inverse of L?

I calculated the inverse of L in R^2 but I don't get it back to the bra-ket-notation. Can somebody help?


BTW: Sorry for my bad english!

 

[Lester]

The inverse of this operator, assuming <u|v>=0, that is orthogonality, is given by 1-|u><v|.

When in doubt about this matter put

$$L^{-1}=a\widehat{1}+b|u><v|$$

and using $$LL^{-1}=L^{-1}L=\widehat{1}$$ fix coeficients a and b. This is a general strategy for this kind of algebraic manipulations.


Jon

 

[Bimmel]

Thx for your fast answer!

I think a had already that kind of inverse operator, but I didn't assume that <u|v>=0. :shy:

Tobi