# If some power $A^n$ of a matrix $A$ is symmetric, is $A$ necessarily symmetric?

If $A^{n}$ is a symmetric matrix, should I conclude that A is also symmetric?

• – Git Gud Jul 30 '15 at 22:22
• Appreciated brothers, its helpful suggestion – Patrick Chidzalo Jul 30 '15 at 22:47

No. Consider the $2 \times 2$ Jordan block $$\pmatrix{0 & 1 \\ 0 & 0},$$ or the matrix $$\pmatrix{0 & -1 \\ 1 & 0},$$ which represents an anticlockwise rotation by $\frac{\pi}{2}$.
• More generally, every asymmetric nilpotent matrix is an example, as is every rotation matrix for an angle a rational, nonintegral multiple of $\pi$. – Travis Willse Jul 31 '15 at 0:01
No you can take the nilpotent matrix $$M=\left(\begin{array}{cccc} 0 & 1 & 0& 0\\ 0&0 & 1& 0\\ 0& 0&0&1\\ 0& 0& 0& 0 \end{array} \right)$$ M isn't symmetrical but $A^4=0_4$ symmetric