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given a symmetric matrix $A = \frac{B+B^T}{2}$ can i retrieve B? Or has that information been lost...

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    $\begingroup$ No: you can't. Information is lost. $\endgroup$ – Crostul Apr 24 '17 at 17:15
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    $\begingroup$ Adding any two anti symmetric matrices will result in zero. $\endgroup$ – copper.hat Apr 24 '17 at 17:15
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    $\begingroup$ “I'm thinking of two numbers whose average is 6. What are they?” Can you see how that's the same question? $\endgroup$ – Matthew Leingang Apr 24 '17 at 17:17
  • $\begingroup$ @MatthewLeingang Yes, but i thought since it was B twise... there was something there. $\endgroup$ – Evan Apr 24 '17 at 17:21
  • $\begingroup$ You're part right, as @OpenBall points out. $\endgroup$ – Matthew Leingang Apr 24 '17 at 17:26
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You can retrieve the diagonal elements of $B$, but not all of $B$. Consider:

$$B = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$$

and

$$B = \begin{bmatrix} 1 & 3 \\ 1 & 1 \end{bmatrix}$$

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