A={{9,7},{-6,-10},{-6,-2}};A//MatrixForm

AtA=Transpose[A].A

Eigensystem[AtA]//MatrixForm

v1={1,1}/Sqrt[2] (* make sure is a unit vector *)
v2={-1,1}/Sqrt[2]

sigma1=Sqrt[288]
sigma2=Sqrt[18]

V=Transpose[{v1,v2}];V//MatrixForm

Sigma={{sigma1,0},{0,sigma2},{0,0}};Sigma//MatrixForm

u1=A.v1/sigma1
u2=A.v2/sigma2

Solve for 3rd column {x,y,z} that is orthogonal to u1 and u2 and has unit length:

Solve[{u1.{x,y,z}==0,u2.{x,y,z}==0,x^2+y^2+z^2==1},{x,y,z}]

Two choices, so pick whichever you prefer for 3rd column of U :

u3={(2/3),(1/3),(2/3)}

U=Transpose[{u1,u2,u3}];U//MatrixForm

U.Sigma.Transpose[V]//MatrixForm (* check that matrices are correct *)