# Any invertible linear map $L$ can be written as $AB$ with $A$ unitary and $B$ psd: a rigorous argument?

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 $L\in (V,V)$?

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