Orthogonality of vectors

Geometrically, one can alternatively take the approach that “orthogonality” is the fundamental structure to be added to a real vector space $${V=\mathbb{R}^{n}}$$. Orthogonality is defined by requiring that:

1. Any subspace $${W}$$ defines an orthogonal complement $${W^{\bot}}$$ such that only the zero vector is contained in both spaces (an orthogonal decomposition)
2. If $${v}$$ is orthogonal to $${w}$$, then $${w}$$ is orthogonal to $${v}$$

One can then look for bilinear forms that vanish for orthogonal vector arguments. A degenerate form features at least one vector $${v}$$ that is orthogonal to every other vector in $${V}$$, thus violating (1) by being in both $${V}$$ and $${V^{\bot}}$$. By linearity, (2) can only be satisfied if the form is either symmetric or anti-symmetric. Thus our only two candidates are a pseudo inner product or a nondegenerate 2-form.

A nondegenerate 2-form is called a symplectic form, and by nondegeneracy can only exist in an even-dimensional vector space, which is then called a symplectic vector space. Every finite-dimensional symplectic vector space is isomorphic to $${\mathbb{R}^{2n}}$$ under the standard symplectic form

$$\displaystyle J=\begin{pmatrix}0 & I\\ -I & 0 \end{pmatrix}$$

(where $${I}$$ is the identity matrix), which is then isomorphic to $${\mathbb{C}^{n}}$$ under the imaginary part of the complex inner product.