# Whether two matrices $M_1$ and $M_2$ are the same, that their entry-wise sums are the same and $M_2=UM_1V$ , where $U$, $V$ are both othogonal

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 $$M_1=M_2$$?

## 1 Answer

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 $A = I\cdot A\cdot I$