# Stuck on singular value decomposition problem

## Homework Statement

Find a singular value decomposition of A.
A^T=
[7 0 5
1 0 5]

## Homework Equations

A = U$$\Sigma$$V^T

## The Attempt at a Solution

I started by doing A^T*A =
[ 74 32
32 26]

Then i went and found the two eigen values lambda1= 90 and lambda2= 10 and the eigenvectors v1 = [2 1]^T and v2 = [-1 2]^T
So, I have V and V^T

From this the singular values are sigma_1 = sqrt(90) and sigma_2 = sqrt(10)
So, $$\Sigma$$ in this decomposition would be
[ sqrt(90) 0
0 sqrt(10)
0 0]

Now to figure out U.
u_1 = 1/sigma_1 AV1 which is
= [ 15/sqrt(90) 0 15/sqrt(90)]^T
and I did the same thing for u_2 to get
[-5/sqrt(10) 0 5/sqrt(10)]

Now, this is where I get stuck. I know I need U to be 3x3 for the matrix multiplication to work out. My book says to find an orthogonal vector and use the gramschmidt method to get u_3. Do I need it to be orthogonal to u_1 or u_2? or both? Also I can't figure out the gramschmidt. If someone could please clarify this for me that would really help. I'm so close (if what I already did is correct) but I can't figure it out.