Quadratic Mathematics | Part IB, 2003

Let UU be a finite-dimensional real vector space endowed with a positive definite inner product. A linear map τ:UU\tau: U \rightarrow U is said to be an orthogonal projection if τ\tau is self-adjoint and τ2=τ\tau^{2}=\tau.

(a) Prove that for every orthogonal projection τ\tau there is an orthogonal decomposition

U=ker(τ)im(τ)U=\operatorname{ker}(\tau) \oplus \operatorname{im}(\tau)

(b) Let ϕ:UU\phi: U \rightarrow U be a linear map. Show that if ϕ2=ϕ\phi^{2}=\phi and ϕϕ=ϕϕ\phi \phi^{*}=\phi^{*} \phi, where ϕ\phi^{*} is the adjoint of ϕ\phi, then ϕ\phi is an orthogonal projection. [You may find it useful to prove first that if ϕϕ=ϕϕ\phi \phi^{*}=\phi^{*} \phi, then ϕ\phi and ϕ\phi^{*} have the same kernel.]

(c) Show that given a subspace WW of UU there exists a unique orthogonal projection τ\tau such that im(τ)=W\operatorname{im}(\tau)=W. If W1W_{1} and W2W_{2} are two subspaces with corresponding orthogonal projections τ1\tau_{1} and τ2\tau_{2}, show that τ2τ1=0\tau_{2} \circ \tau_{1}=0 if and only if W1W_{1} is orthogonal to W2W_{2}.

(d) Let ϕ:UU\phi: U \rightarrow U be a linear map satisfying ϕ2=ϕ\phi^{2}=\phi. Prove that one can define a positive definite inner product on UU such that ϕ\phi becomes an orthogonal projection.

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