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 \({J_{\mu\nu}\equiv J\left(e_{\mu},e_{\nu}\right)}\) and \({I}\) is the identity matrix), which is then isomorphic to \({\mathbb{C}^{n}}\) under the imaginary part of the complex inner product.

An Illustrated Handbook