No idea how to prove this property about symmetric matrices This is from homework, so please hints only.

Suppose $A$ is symmetric such that all of its eigenvalues are 1 or -1. Prove that $A$ is orthogonal.

The converse is really easy, but I really have no idea how to do this. Any hints?

Edited Saturday 16 November 2013 10:03 PM PST
Well, it seems the "hints" have had their desired effect, so I'm editing this post to be an answer, pure and simple.
That being said, try this:
since $A$ is symmetric, there exists orthogonal $O$ such that $O^TAO = \Lambda$, with $\Lambda$ diagonal and $\Lambda_{ii} = \pm 1$ for all $i$.  Then $\Lambda^T\Lambda = I$ and since $O^TA^TO = \Lambda^T$, we have $O^TA^TOO^TAO = O^TA^TAO = \Lambda^T\Lambda$ = I.
Thus $A^TA = OIO^T = OO^T = I$ and $A$ is orthogonal. QED
Whew!  That feels better!
Well, now I've certainly said too much! ;- )!!!
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
