singular value decomposition Given regular matrices $A_i,B_i \in \textrm{GL}_n(\mathbb{R}),$ $i=1,2$.
Let $A_1 = U_1 B_1 V_1$ and $A_2=U_2 B_2 V_2$ where $U_i,V_i \in \textrm{O}_n(\mathbb{R})$ $(i=1,2)$ are orthogonal matrices. This means that $A_1$ and $B_1$ resp. $A_2$ and $B_2$ have the same singular values.
Now let $A_2 A_1^{-1}$ and $B_2 B_1^{-1}$ have the same singular values.
I assume that in this case one can find $U_i',V' \in \textrm{O}_n(\mathbb{R})$ so that $A_1 = U_1' B_1 V$ and $A_2=U_2' B_2 V$.
Is this true, at least under some conditions? Or is there a simple counterexample?
 A: Let $A_2A_1^{-1}=U_A DV_A$ and $B_2B_1^{-1}=U_B DV_B$ be the two SVDs with the same $D$.
Set $U_2'=U_AU_B'$, $U_1=V_B'V_A$, $V=B_1^{-1} U_1 A_1$.
The first equality is clear. The second one is proved by $$U_2'B_2V=U_A U_B'B_2 B_1^{-1} V_B' V_A A_1=U_ADV_A A_1=A_2A_1^{-1}A_1=A_2.$$
A: This is really a commentary, but too loong ...
I dont really understand. ¿Is the $B_i$ supposed to be diagonal, as it looks like, since the title is the "singuar value decomposition"?
But then $B_i$ has the singular values of $A_i$.  Assuming this, both $A_1^T A_1$ and $A_2^T A_2$ are symmetric positive definite matrices, so by simultaneous diagonalization there exists a diagonal matrix $\Lambda$ with positive entries and a non-singular matrix (not orthogonal!) so that $X^T A_1^T A_1 X=\Lambda$ and $X^T A_2^T A_2 X=I$.   It follows that $(A_2 X)^T(A_2 X)=I$ so $A_2 X$ is orthogonal, write $A_2 X=V$. We have $A_1^T A_1 = X^{-T} \Lambda X^{-1} = (\Lambda^{1/2} X^{-1})^T \Lambda^{1/2} X^{-1}$ so we can conclude that $\Lambda^{1/2}X^{-1}$ is one square root of   $A_1^T A_1$.  $A_1$ is another square root, and since all square roots are orthogonally related, there exist some orthogonal matrix $U$ such that $A_1 = U \Lambda^{1/2} X^{-1}$ and we have already found there is some orthogonal matrix $V$ such that $A_2 = V X^{-1}$. This is close to what you have given, but the common matrix in the two expressions is not orthogonal, but general non-singular. 
