Why not different diagonal elements in orthogonal reduction of quadratic form?

Question

Suppose we have quadratic form $(3x)^2 -(2y)^2-z^2-(4xy)+(8xz)+(12yz)$

and we are asked to reduce it to cannonical form. The steps would be:

1. Find eigen values (3,6,-9 in this case) of the matrix of quadratic form (A)
2. Modal matrix(P) from eigen vectors
3. Normalisation of P.(N)
4. Finding $D=N^T×A×N$

D would look like

$$D =\begin{bmatrix} 3 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0& -9 \end{bmatrix}.$$

Now the diagonal elements are exactly same as eigen values! My query is till now, every time, eigen values come out as the diagonal elements, then why we go about finding the matrix in the first place?