From eigenvectors onwards

I have found the the three eigenvectors(i hope) which are:

v1=[1;0;-1] v2=[1;1;1] v3=[-1;2;-1]

I have used the following method on them: (<-- is the pivot row)

1 0 -1 <--

1 1 1

__-1 2 -1__

0 1 2 <--

__0 2 -2__

0 0 -6<--

Which, when the pivot rows are written as columns of a U, gives U as being:[1,0,0;0,1,0;-1,2,-6]

Which when applied to (U^t)*M*U this matrix, U, finds D to be:[6,-6,-18;-6,18,-42;-18,-42,108]

Which satisfies the question "Find a matrix(any matrix) U, so that (U^t)*M*U is diagonal"....correct?

Is this a valid method for working out a matrix, U?