16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10

Inner Product Spaces

Mathematics — Linear Algebra · rank-one-operators

Problem

Let VV be a finite-dimensional inner product space. For each α,β\alpha, \beta in VV, let Tα,βT_{\alpha,\beta} be the linear operator on VV defined by Tα,β(γ)=(γ∣β)αT_{\alpha,\beta}(\gamma) = (\gamma|\beta)\alpha. Show that (a) Tα,β∗=Tβ,αT^*_{\alpha,\beta} = T_{\beta,\alpha}. (b) trace(Tα,β)=(α∣β)\text{trace}(T_{\alpha,\beta}) = (\alpha|\beta). (c) Tα,βTγ,δ=Tα,(β∣γ)δT_{\alpha,\beta}T_{\gamma,\delta} = T_{\alpha,(\beta|\gamma)\delta}. (d) Under what conditions is Tα,βT_{\alpha,\beta} self-adjoint?

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