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Given two matrices $M_1,M_2\in R^{k\times l}$, and there exist two orthogonal matrices $U\in R^{k\times k}$, $V\in R^{l\times l}$ such that $M_2=UM_1V$. If we know the sum of elements of $M_1$ and $M_2$ are the same, $$\sum\limits_{i,j}m^{(1)}_{i,j}=\sum\limits_{i,j}m^{(2)}_{i,j},$$ where $m^{(1)}$ are the elements of $M_1$ and $m^{(2)}$ are the elements of $M_2$. Then can we conclude that $M_1=M_2$?

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no we can't: let $A=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ and $B=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ with $a,b,c,d\in \mathbb{R}$

$A\cdot A^T = I \Rightarrow$ A is orthogonal

$A\cdot B \cdot A^T =\begin{pmatrix}d&c\\b&a\end{pmatrix}$

Therefor the sum is the same

qed

edit: yes the statement 'can imply' is always true... you can simply take I $$I\cdot I^T = I \Rightarrow I $$ is orthogonal and $A = I\cdot A\cdot I$

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