Quadratic Mathematics | Part IB, 2002

Define the adjoint α\alpha^{*} of an endomorphism α\alpha of a complex inner-product space VV. Show that if WW is a subspace of VV, then α(W)W\alpha(W) \subseteq W if and only if α(W)W\alpha^{*}\left(W^{\perp}\right) \subseteq W^{\perp}.

An endomorphism of a complex inner-product space is said to be normal if it commutes with its adjoint. Prove the following facts about a normal endomorphism α\alpha of a finite-dimensional space VV.

(i) α\alpha and α\alpha^{*} have the same kernel.

(ii) α\alpha and α\alpha^{*} have the same eigenvectors, with complex conjugate eigenvalues.

(iii) If Eλ={xV:α(x)=λx}E_{\lambda}=\{x \in V: \alpha(x)=\lambda x\}, then α(Eλ)Eλ\alpha\left(E_{\lambda}^{\perp}\right) \subseteq E_{\lambda}^{\perp}.

(iv) There is an orthonormal basis of VV consisting of eigenvectors of α\alpha.

Deduce that an endomorphism α\alpha is normal if and only if it can be written as a product βγ\beta \gamma, where β\beta is Hermitian, γ\gamma is unitary and β\beta and γ\gamma commute with each other. [Hint: Given α\alpha, define β\beta and γ\gamma in terms of their effect on the basis constructed in (iv).]

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