Consider the problem of finding the minimum and maximum of the quadratic form $$r(\textbf{x})=x_1^2+x_2^2+x_3^2$$ subject to the constraint $q(\textbf{x})=x_1^2+3x_2^2+x_3^2+2x_1x_2-2x_1x_3-2x_2x_3=1$.
By finding the eigenvalues of the matrix of $q$, the diagonal representation is $q$ is $y_2^2+4y_3^2$. That of $r$ stays the same, namely $y_1'^2+y_2'^2+y_3'^2$. One could solve the problem by inserting one of the diagonal representations inside the other and minimise that expression.
Why is this possible? Aren't the diagonal representations based on two different basis, i.e. the basis of eigenvectors of $q$ and $r$ respectively, and so represent different coordinates, as the notation also indicates?
As a bonus question. The minimum and maximum values of a quadratic form, with ascending eigenvalues indiced from $1$ to $n$, is given by:
$$\lambda_1={\text{min}} \ \frac{r(\textbf{x})}{\|\mathbf x\|^2} \quad \lambda_n={\text{max}} \ \frac{r(\textbf{x})}{\|\mathbf x\|^2}$$
How would one solve the problem using these definitions?
The theorem you have probably used is the following: Let $(E, r)$ be an Euclidean space, and let $q:E\to \mathbb{R}$ be a quadratic form. Then there is an orthonormal basis of $E$ (for its scalar product) which is $q$-orthogonal (that is, the matrix of $q$ is this basis is diagonal).
What does it means concretely? If $(e_1,\ldots,e_n)$ is such a basis, write ${\bf x}=y_1e_1+...+y_ne_n$ (with your notation). Since the basis is orthonormal for $r$, we have $r({\bf x})=y_1^2+...+y_n^2$. Since the basis is $q$-orthogonal, you have $q({\bf x})=a_1y_1^2+...+a_ny_n^2$ for some real numbers $a_1,...,a_n$.
Let me insist on the fact that $y_1,...,y_n$ appear in both $q$ and $r$; these are the coordinates on the same vector ${\bf x}$ in your new basis.
The way to compute such a basis is as follows. To simplify, I only consider the case $E=\mathbb{R}^n$ with its standard scalar product. Write the matrix $M$ of $q$ in the canonical basis. Then an orthonormal basis $B$ of $\mathbb{R}^n$ consisting of eigenvectors of $M$ is the basis you seek.
That's what you have probably done, and that's why you have the result you are talking about.