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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 definite matrix, 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 $E$ and called the symplectic spectrum.

Now this theorem is shown as Lemma 2.43 in the book "Introduction to symplectic topology" by McDuff and Salamon. In the proof, they construct such $r_1,..,r_n$ and I understand the steps in the proof. But at one point they say: Define $\Delta(r) = \text{diag}(\frac{1}{r_1^2},...,\frac{1}{r_n^2},\frac{1}{r_1^2},...,\frac{1}{r_n^2})$, then in order to show uniqueness of the $r_1,...,r_n$, it is sufficient to show that from $\psi^T \Delta (r) \psi = \Delta (r').$ it follows that $r=r'.$

Now, I don't see how this is related to the uniqueness stated in the theorem.

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