{vals,vecs}=Eigensystem[(1/2)(({{1, 0, 1}, {0, 1, 0}, {1, 0, 1}}))]

vecs=Orthogonalize[vecs]

reconstruct the matrix:

A=Outer[Times,{(1/(Sqrt[2])),0,(1/(Sqrt[2]))},{(1/(Sqrt[2])),0,(1/(Sqrt[2]))}]+((1/2))Outer[Times,{0,1,0},{0,1,0}] +0

(* this would be prettier if instead we wrote: *)
A= Sum[vals[[i]] Outer[Times, vecs[[i]],vecs[[i]]], {i,1,3}]

MatrixForm[A]

check that the eigenvectors resolve the identity:

Sum[Outer[Times, vecs[[i]],vecs[[i]]], {i,1,3}]