Please teach me how to argue for this rigorously: The question:
$L$ is an invertible linear map with $L\in (V,V)$. Prove that $L$ can be written as $AB$ where $A^\dagger A=I=AA^\dagger$ and $B$ is positive-definite.
What I think:
This is intuitively true because $L^\dagger L$ is positive-definite. So $L^\dagger L=B^\dagger B$ for some positive definite $B$ which does not necessarily equal $L$. To correct for that, we throw in $A$ which cancels.
I don't know how to present this rigorously though. Thanks in advance.
Added: Curious, what can I say about a general $L\in (V,V)$?