I need to prove/disprove that if A^2 is orthogonal matrix, than A is also orthogonal.
I think the statement is true, because if A^2 orthogonal, then A^tA^t = A^-1A^-1, hence AAA^tA^t = AAA^-1A^-1 = AIA^-1 = AA^-1 = I, hence A is orthogonal.
Is my proof right?
You're going to have a hard time proving this, because it's not true.
Consider the matrix:
$A = \begin{bmatrix}1&2\\-1&-1\end{bmatrix}$
Now $A^2 = -I$ and $-I$ is an orthogonal matrix:
$(-I)^T = -I$ and $(-I)(-I)^T = (-I)(-I) = I$
However, $A$ is not orthogonal.
Does X^{2}=A^{2} → X=±A?
No, because matrices have divisors of 0. Explanation:
X^{2}-A^{2}=(X-A)(X+A)=0 does not imply X-A=0 or X+A=0 for matrices.
If you prefer a counter example, that allegedly has been done for a 2X2 matrix (I tend not to work through Deveno’s posts, but I’ll take romseks’ word for it).
OK, I’m happy. My question is answered. Since everyone else was apparently happy before my posts, we are all happy now.
EDIT: In answer to the OP, one could say if A^{2} is orthogonal, A could be orthogonal, but not necessarily.