I have a question about a step in the proof of the following theorem from symplectic geometry:
The theorem is: Given any ellipsoid $E:=\{ w \in \mathbb{R}^{2n}; \sum_{i,j =1}^{2n} a_{ij}w_iw_j \le 1\}$ with $(a_{ij})$ being a symmetric and positive definiete, there is a symplectic linear transform $\psi \in Sp(2n) $ such that $\psi(E)=\{z \in \mathbb{C}^{n}; \sum_{i=1}^{n} \vert\frac{z_i^2}{r_i^2} \vert \le 1\}$ for some n-tuple $r=(r_1,...,r_n)$ with $0 < r_1 \le ...\le r_n$. Moreover, the numbers $r_j$ are uniquely determined by $E.$
Now this theorem is shown as Lemma 2.43 in the book Introduction to symplectic topology by McDuff and Salamon and they construct such $r_1,..,r_n$ and I get a steps in the proof but then say: Define $\Delta(r) = \text{diag}(\frac{1}{r_1^2},...,\frac{1}{r_n^2},\frac{1}{r_1^2},...,\frac{1}{r_n^2})$ and then for uniqueness it is sufficient to show that from $\psi^T \Delta (r) \psi = \Delta (r').$ it follows that $r=r'.$
Now, I don't see why this is sufficient to uniqueness.
In their proof, McDuff an Salamon use the symbol $\Psi$ to denote two different (but related) symplectic matrices ; This may be the source of your confusion. In any case, here is a more detailed explanation of this step.
There is a one-to-one correspondence between symmetric positive-definite matrices and non-degenerate ellipsoids given by
$$ S = (S_{ij}) \; \leftrightarrow \; E_S := \{ x \in \mathbb{R}^n : x^T S x \le 1 \} \, .$$
If $L : \mathbb{R}^n \to \mathbb{R}^n$ is a linear isomorphism, then we have
$$ L(E_S) = \{ Lx \in \mathbb{R}^n : x^T S x \le 1 \} = \{ y \in \mathbb{R}^n : y^T (L^{-1})^{T}SL^{-1} y \le 1 \} = E_{(L^{-1})^{T}SL^{-1}} \, .$$
Hence, $L$ brings $E_S$ to a 'standard' ellipsoid $E(r) = \{ x \in \mathbb{R}^n : \sum_{i=1}^n \frac{x_i^2}{r_i^2} \le 1 \} $ if and only if $(L^{-1})^{T}SL^{-1} = \mathrm{diag} \left( r_1^{-2}, \dots, r_n^{-2} \right)$.
Consider a symmetric positive-definite matrix $A = (a_{ij}) \in \mathbb{R}^{2n \times 2n}$ ; One can show that the corresponding 'symplectic ellipsoid' $E = \{ w \in \mathbb{R}^{2n} : w^TAw \le 1 \}$ can be sent to a 'standard symplectic ellipsoid' $E(r,r) = \{ w \in \mathbb{R}^{2n} : w^T\Delta(r)w \le 1 \}$ by some symplectic matrix $\Psi \in \mathrm{Sp}(2n)$.
Now, if some other symplectic matrix $\Phi \in \mathrm{Sp}(2n)$ sends $E$ to another 'standard symplectic ellipsoid' $E(r', r')$, you want to show that $r'$ is a permutation of $r$. Relying on what we established above, we have
$$ \Phi^T \Delta(r') \Phi = A = \Psi^T \Delta(r) \Psi \, .$$
Since the set of symplectic matrices is a group, $\Theta := \Psi \Phi^{-1} \in \mathrm{Sp}(2n)$ and we have $\Delta(r') = \Theta^T \Delta(r) \Theta$. In their book, McDuff and Salamon show that this relation implies that $r$ and $r'$ differ by a permutation.