Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose we are given two arbitrary $m \times n$ matrices, $A$, $B$, where we know $B$ has full column rank. Let $m>>n$. Can we always find a square $m \times m $ matrix $X$, such that $A=XB$? I do not care if $X$ is unique or not, as long as one exists.

share|improve this question
    
In optimization, can we optimize wrt $X$ instead of $A$? –  user25004 Oct 18 '12 at 5:58
add comment

1 Answer

up vote 2 down vote accepted

Yes. The rank of $B$ is $n$, so $B$ has a trivial nullspace. Take $X = AB^{+}$, where $B^{+}$ is the Moore-Penrose pseudoinverse of $B$. Then $$XB = AB^{+}B = AI_n = A.$$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.