Is every matrix in $M_n(\mathbb{C})$ a sum of symplectic matrices?

Question

A matrix $A \in GL(2n)$ is called symplectic iff $A^{t}JA = J$ where $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$. Is it true that every matrix in $M_{2n}(\mathbb{C})$ can be written as a sum of symplectic matrices? Any hints on how to prove this?

Answer 1

The result is indeed proved in the paper Every $2n$-by-$2n$ complex matrix is a sum of three symplectic matrices, in Linear Algebra and its Applications no. 517 (2017). For $n=1$ one can improve the result, i..e., that every complex matrix of size $2$ is the sum of two symplectic matrices. There are examples, which show that some matrices need at least three summands. So the fatcor $3$ is best possible in general.