I have a got a matrix $M$, is it always possible to find a SYMMETRIC matrix $S$ such that its square equals $M$, namely

$S S = M$

and if the answer is positive, what is the procedure to calculate this symmetric matrix $S$?

  • $\begingroup$ The matrix $\left(\begin{matrix} 0&1 \\ 0&0 \end{matrix}\right)$ is never a square (because if it was a square of some matrix $A$, then $A$ would satisfy $A^4=0$, and thus $A^2=0$ by a well-known property of nilpotent matrices). This works over any field. $\endgroup$ Jun 11, 2015 at 17:18
  • $\begingroup$ If $S$ is symmetric then $M$ as well... $\endgroup$
    – daw
    Jun 11, 2015 at 18:36
  • $\begingroup$ ok @daw so let us suppose $M$ is symmetric. How would you find $S$? thanks! $\endgroup$
    – johnhenry
    Jun 11, 2015 at 22:41


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