We start by finding the singular values of , to do this we need to choose if we want to find the eigenvalues of or . In this case makes sense to choose since this results in us only needing to find eigenvalues. Computing the eigenvalues and singular values we get the following:
From here we can construct our matrix (ensuring it is ) to fit the full SVD definition:
Next we can find the eigenvectors which will make up the columns of .
This gives us a final matrix for :
Next we calculate and using the formula this results in:
This would be enough for thin SVD but we are looking for the full SVD so to get we use the following equation using this we get:
So our final matrix for is:
From here we can write the full SVD: