# what can we say about the SVD of a matrix with respect to another SVD?

I have two matrices, both are $n \times m$ where $m < n$. These two matrices are $A$ and $B$.

I also know the singular value decomposition of $AB^{\top} = U \Sigma V^{\top}$.

Is there anything I can say about the SVD of $A$ or $B$? (for example, can I say that the left singular vectors of $A$ are $U$?) Are there any conditions under which I could say something like that?

Thanks.

• Nothing else special about either matrix? – J. M. is a poor mathematician Jul 23 '12 at 18:27
• nothing really, other than perhaps full rank ($m$), if that can help... but I prefer to avoid making that assumption. – kloop Jul 23 '12 at 18:29
• but I would like to hear if there are any assumptions under which some properties of the SVD of $A$ could be identified. – kloop Jul 23 '12 at 18:43

Let $V_A$ be the right singular vectors of $A$, and $U_B$ be the left singular vectors of $B^T$, we have $$AV_A=U_A\Sigma_A, \quad B^TV_B=U_B\Sigma_B$$
If $V_A= U_B$, then $$AB^TV_B=AU_B\Sigma_B=AV_A\Sigma_B=U_A\Sigma_A\Sigma_B$$
Also, because the the left singular vectors of $B^T$ are right singular vectors of $B$, we can see that, if the right singular vectors of $B$ are right singular vectors of $A$, then we have that the right singular vectors of $B^T$ is the right singular vectors of $AB^T$, and the left singular vectors of $A$ is the left singular vectors of $AB^T$.