Define the adjoint of an endomorphism of a complex inner-product space . Show that if is a subspace of , then if and only if .
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 of a finite-dimensional space .
(i) and have the same kernel.
(ii) and have the same eigenvectors, with complex conjugate eigenvalues.
(iii) If , then .
(iv) There is an orthonormal basis of consisting of eigenvectors of .
Deduce that an endomorphism is normal if and only if it can be written as a product , where is Hermitian, is unitary and and commute with each other. [Hint: Given , define and in terms of their effect on the basis constructed in (iv).]