| Abstract: |
| By embedding the real symplectic group $\text{Sp}(2n)$ in $\mathbb{R}^{4n^2}$, we construct the gradient embedded vector field, which is the gradient vector field of a cost functional defined on the symplectic group, and we write it in the ambient coordinates. We present necessary and sufficient conditions for critical points of the cost functional. We further give an explicit formula for the Hessian operator on the symplectic group written in the ambient coordinates. As an application, we explicitly describe the steepest descent and Newton algorithms on $\text{Sp}(2n)$. |
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