2.II.17F

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

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 $\alpha$ of a finite-dimensional space $V$.

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

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

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

(iv) There is an orthonormal basis of $V$ consisting of eigenvectors of $\alpha$.

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

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