linear complex structure

A on a real vector space $V$ , with $\\mathrm{dim}(V)=m$ , is a linear automorphism $J\\in\\mathrm{Aut}(V)$ such that $J^{2}=J\\circ J=-\\mathrm{id}_{V}$ . With a complex structure $J$ we can consider $V$ as a complex vector space with the product $\\mathbb{C}\\times V\\rightarrow V$ given by

(x+iy)\\mathbf{v}=x\\mathbf{v}+yJ(\\mathbf{v}),\\ \\ \\forall x,y\\in\\mathbb{R},\\ %\\mathbf{v}\\in V.

This implies that the dimension $m$ of $V$ must be even.

A common example is $V=\\mathbb{R}^{2n}$ with the standard basis $\\mathbf{e}_{1},...,\\mathbf{e}_{n},\\mathbf{f}_{1},...,\\mathbf{f}_{n}$ , for which we can obtain a complex structure $J_{0}\\in\\mathrm{Aut}(\\mathbb{R}^{2n})$ represented by the matrix

\\left(\\begin{array}[]{cc}\\mathbf{0}&\\mathbf{I}_{n}\\\\-\\mathbf{I}_{n}&\\mathbf{0}\\end{array}\\right).

Here $\\mathbf{I}_{n}\\in\\mathrm{M}_{n}(\\mathbb{R})$ is the identity $n\\times n$ matrix and $\\mathbf{0}\\in\\mathrm{M}_{n}(\\mathbb{R})$ is the zero $n\\times n$ matrix.