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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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1 Answer 1

up vote 2 down vote accepted

Use singular value decomposition if you are working with finite dimensional real or complex vector spaces.

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The same method answers the "Added" question, showing that for singular matrices you just replace "positive definite" with "positive semidefinite". –  Noah Stein Dec 3 '11 at 23:12
    
Thanks a lot, guys! –  Nathan Dec 4 '11 at 22:21

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