Calculating the Inverse of Operator L in R^2

In summary, the conversation discusses the operator L=\widehat{1}+|u><v| and finding its inverse. It is mentioned that the inverse is 1-|u><v|, assuming <u|v>=0. A general strategy for finding such inverse operators is also mentioned.
  • #1
Bimmel
2
0
hello,

given is the Operator L=[tex]\widehat{1}[/tex]+|u><v|, where [tex]\widehat{1}[/tex] 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!
 
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  • #2
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

[tex]L^{-1}=a\widehat{1}+b|u><v|[/tex]

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


Jon
 
Last edited:
  • #3
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
 

1. What is an operator in R^2?

An operator in R^2 is a mathematical function that operates on a vector in the two-dimensional real coordinate space. It can perform various operations such as translation, rotation, scaling, etc.

2. What is the inverse of an operator in R^2?

The inverse of an operator in R^2 is a mathematical function that reverses the effects of the original operator. It essentially "undoes" the transformations performed by the original operator.

3. Why is it important to calculate the inverse of an operator in R^2?

Calculating the inverse of an operator in R^2 is important because it allows us to undo the transformations performed by the operator, making it easier to manipulate and analyze the original data. It also allows us to solve equations involving the operator and its inverse.

4. How do you calculate the inverse of operator L in R^2?

The inverse of operator L in R^2 can be calculated by using the inverse matrix method. This involves finding the inverse of the matrix representation of the operator, which can then be used to create the inverse operator.

5. What are some applications of calculating the inverse of operator L in R^2?

The inverse of operator L in R^2 has various applications in mathematics, physics, and engineering. It is used in solving differential equations, optimization problems, and in the analysis of linear systems. It is also commonly used in computer graphics and image processing to manipulate and transform images.

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