Example. An elementary skew matrix [meinrenken2013clifford, Section 2.2.10, Proposition 2.12, p. 40] [fcap-000S]
Example. An elementary skew matrix [meinrenken2013clifford, Section 2.2.10, Proposition 2.12, p. 40] [fcap-000S]
This coordinate example specializes TauCeti's extension of the characteristic-zero construction summarized in Convention [fcap-000N]. Take \(V=R^n\) with the standard sum-of-squares quadratic form, and let \(e_i,e_j\) be distinct coordinate vectors. Under the standard-coordinate realization, \[e_i\wedge e_j\longmapsto 2(E_{ij}-E_{ji}).\] Thus its action on \(x=(x_k)\) is \[x\longmapsto 2x_j e_i-2x_i e_j.\] The factor \(2\) is the polar-form normalization from Convention [fcap-000N]: for the sum-of-squares form, \(B_Q(x,y)=2\sum _kx_ky_k\). The displayed matrix is the standard-coordinate specialization of the basis-free isomorphism in Meinrenken's proposition. Dividing the skew matrix by two would correspond to using the associated bilinear form \(B_Q/2\) instead.