@@ -255,7 +255,7 @@ which requires that $x_t' R x_t$ converge to zero as $t \rightarrow + \infty$.
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255 | 255 | |
256 | 256 | +++ |
257 | 257 | |
258 | | -## Reciprocal pairs property |
| 258 | +## Reciprocal Pairs Property |
259 | 259 | |
260 | 260 | To proceed, we study properties of the $(2n \times 2n)$ matrix $M$ defined in {eq}`Mdefn`. |
261 | 261 | |
@@ -273,7 +273,12 @@ $$
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273 | 273 | MJM^\prime = J. |
274 | 274 | $$ (eq3) |
275 | 275 | |
276 | | -It can be verified directly that $M$ in equation is symplectic. |
| 276 | +Salient properties of symplectic matrices that are readily verified include: |
| 277 | + |
| 278 | + * If $M$ is symplectic, then $M^2$ is symplectic |
| 279 | + * The determinant of a symplectic, then $\textrm{det}(M) = 1$ |
| 280 | + |
| 281 | +It can be verified directly that $M$ in equation {eq}`Mdefn` is symplectic. |
277 | 282 | |
278 | 283 | It follows from equation {eq}`eq3` and from the fact $J^{-1} = J^\prime = -J$ that for any symplectic |
279 | 284 | matrix $M$, |
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