I am trying to construct a Unitary matrix U (unitary: U*U=UU*=In) from the eigen

Question

I am trying to construct a Unitary matrix U (unitary: U*U=UU*=In) from the eigen vectors of this matrix A:
A =
1 2 2 2
2 1 2 2
2 2 1 2
2 2 1 2
My characteristic polynomial being(y=lambda): y^4-4y^3-18y^2-20y-7=(y-7)(y+1)^3
My eigen values are correct being: 7,-1,-1,-1
My eigen values, according to most "calculators" and humans, are (respectively):
v1=(1,1,1,1) v2=(-1,0,0,1) v3=(-1 0 1 0) v4=(-1 1 0 0)

I have been lead to believe that if you normalize these vectors and form a matrix from them you will get a unitary matrix U st U*U=UU*=In BUT! this is not the case and I am at a loss.

Can some one please show me how to construct a Unitary matrix from this information?
I am either encountering 3 problems:
1. My eigen vectors are incorrect (which is unlikely)
2. I am not normalizing properly (which I believe is how you get the unitary)
3. I am entirely mistaken about how to form a unitary matrix.
PLEASE SHOW THE PROCESS I HAVE BEEN POUNDING MY HEAD AGAINST THIS PROBLEM FOR HOURS AND HOURS AND HOURS...and I am dumbfounded as to why it won't work :(

Explanation / Answer

Yes, the vectors verify but you have to make the vectors orthonormal
(by Gram-Schmidt orthogonalization process)

as it says here: http://en.wikipedia.org/wiki/Spectral_theorem#Normal_matrices

v1 and v2 are orthogonal already
So the new vectors ae
u1=1/2(1,1,1,1)
u2=1/2 (-1,0,0,1)

I calculated u3=1/6(-1,0,2,-1)

So u4 should be

u4=v4-(v4*u1)u1-(v4*u2)u2-(v4*u3)u3 , where * is the dot(scalar) product

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