Proving Matrix Equality Using Singular Value Decomposition

[linearishard]

Hi, I have another question, if A and B are mxn matrices, how do I prove that $AA^T = BB^T$ iff $A = BO$ where $O$ is some orthogonal matrix? I think I need to use a singular value decomposition but I am not sure. Thanks!

 

[Euge]

Can you at least prove the reverse direction, that is, if $A = BO$ for some orthogonal matrix $O$, then $AA^T = BB^T$? You don't need to use SVD for this.

 

[linearishard]

Yeah I did that but it seemed too simple, my study guide says I should be using SVD. Is it actually unnecessary?

 

[Euge]

You use SVD for the forward direction, not the reverse direction.

 

[linearishard]

what do you mean by that? What is the forward and reverse directions?

 

[Euge]

The forward direction: If $AA^T = BB^T$, then $A = BO$ where $O$ is some orthogonal matrix. The reverse direction: If $A = BO$ where $O$ is an orthogonal matrix, then $AA^T = BB^T$.

 

[Euge]

If $AA^T = BB^T$, then the singular values of $A$ are the singular values of $B$. You can write $A = U\Sigma V^T$ and $B = U\Sigma Q^T$ for some orthogonal matrices $U, V$, and $Q$. Then $A = BO$ where $O = QV^T$. Since the transpose and product of orthogonal matrices are orthogonal, $O$ is orthogonal.