Kunze Linear Algebra · Problem 10.3.5

Bilinear Forms

Mathematics — Methods · skew-symmetric-projection-operator

Problem

Let VV be an nn-dimensional vector space over a subfield FF of CC. Prove the following. (a) The equation (Pf)(α,β)=12f(α,β)−12f(β,α)(Pf)(\alpha, \beta) = \frac{1}{2}f(\alpha, \beta) - \frac{1}{2}f(\beta, \alpha) defines a linear operator PP on L(V,V,F)L(V, V, F). (b) P2=PP^2 = P, i.e., PP is a projection. (c) rank P=n(n−1)2P = \dfrac{n(n-1)}{2}; nullity P=n(n+1)2P = \dfrac{n(n+1)}{2}. (d) If UU is a linear operator on VV, the equation (U†f)(α,β)=f(Uα,Uβ)(U^\dagger f)(\alpha, \beta) = f(U\alpha, U\beta) defines a linear operator U†U^\dagger on L(V,V,F)L(V, V, F). (e) For every linear operator UU, the projection PP commutes with U†U^\dagger.

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