2.3. SYMPLECTIC MAPPINGS 17 Proof. Let us compute dη ∧ dy = (Cdx + Ddξ) ∧ (Adx + Bdξ) = AT Cdx ∧ dx + BT Ddξ ∧ dξ + (AT D − CT B)dξ ∧ dx = dξ ∧ dx if and only if (2.3.4) AT C and BT D are symmetric, AT D − CT B = I. Furthermore, KT JK = AT CT BT DT O I −I O A B C D = AT C − CT A AT D − CT B BT C − DT A BT D − DT B = J if and only if (2.3.4) holds. We record some useful observations: THEOREM 2.4 (More on symplectic matrices). (i) The product of two symplectic matrices is symplectic. (ii) If K is a symplectic matrix, then (2.3.5) σ(Kz, Kw) = σ(z, w) (z, w ∈ R2n). (iii) A matrix K is symplectic if and only if (2.3.6) K is invertible, K−1 = JKT JT . (iv) If AT J + JA = 0, then Kt := exp(tA) is symplectic for each t ∈ R. Proof. Assertions (i), (ii), and (iii) follow directly from the definitions and the fact that JT = −J = J−1. To prove (iv), write Wt := KT t JKt − J and compute ∂tWt = AT Wt + WtA + AT J + JA = AT Wt + WtA. Since W0 = 0, we deduce from uniqueness that Wt = 0 for all t ∈ R.

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