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Suppose we have an $n\times m$ matrix $M$ of rank $k\ll n$. Such a matrix can be decomposed $$ M = A B $$ where $A$ is an $n\times k$ matrix and $B$ is a $k\times m$ matrix. Such a decomposition can be obtained for example using (compact) singular value decomposition $$ M = U\Sigma V^* \\ A=U , B=\Sigma V^*$$

Clearly, this decomposition is not unique. For example, for any number $x$ $$ A=U \Sigma^{1-x}, B=\Sigma^x V^*$$ will also have $A B=M$

Is there a way of characterising or parametrizing the space of all matrices $A$ and $B$ (of known size $n\times k$ and $k\times m$ respectively) whose product is a given matrix $M$?

Wouter
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  • For any $k\times k$ invertible matrix C we still have $ (U C) (C^{-1}\Sigma V^*) = M$. This might be the whole set. – Wouter Dec 16 '15 at 05:27

1 Answers1

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You are right. Assume that $M=AB$ where $A:K^k\rightarrow K^n$ is one to one and $im(A)=im(M)$; note that $U:K^k\rightarrow K^n$ has same two properties. There is a unique $C:K^k\rightarrow K^k$ s.t. $A=UC$ and $C$ is a bijection. Indeed, $C$ is defined by $C(x)=U^{-1}(A(x))$.