Grand
Mar14-11, 05:28 PM
1. The problem statement, all variables and given/known data
Theorem
If, for given A, both operators A_l^{-1} and A_r^{-1} exist, they are unique and
A_l^{-1}=A_r^{-1}
The proof is rather straightforward, at least the first part of it:
A_l^{-1}A=I/\leftarrow A_r^{-1}
A_l^{-1}AA_r^{-1}=A_r^{-1} (1)
A_l^{-1}A=I/\rightarrow A_l^{-1}
A_l^{-1}AA_r^{-1}=A_l^{-1} (2)
Therefore
A_l^{-1}=A_r^{-1}
However, then they say that this proof holds for any pair of operators A_l^{-1} and A_r^{-1} (which I can't deny) and that eqs (1) and (2) ensure there exists only one such pair, which I can't understand. I would be very grateful if someone explains to me why is that.
Theorem
If, for given A, both operators A_l^{-1} and A_r^{-1} exist, they are unique and
A_l^{-1}=A_r^{-1}
The proof is rather straightforward, at least the first part of it:
A_l^{-1}A=I/\leftarrow A_r^{-1}
A_l^{-1}AA_r^{-1}=A_r^{-1} (1)
A_l^{-1}A=I/\rightarrow A_l^{-1}
A_l^{-1}AA_r^{-1}=A_l^{-1} (2)
Therefore
A_l^{-1}=A_r^{-1}
However, then they say that this proof holds for any pair of operators A_l^{-1} and A_r^{-1} (which I can't deny) and that eqs (1) and (2) ensure there exists only one such pair, which I can't understand. I would be very grateful if someone explains to me why is that.