Exercise MM.T35 asked you to show that $A\adjoint{A}$ is Hermitian. Prove directly that $A\adjoint{A}$ is a normal matrix.

In the discussion following Theorem OBNM we comment that the equation $\hat{\vect{x}} = \adjoint{U}\vect{x}$ is just Theorem COB in disguise. Formulate this observation more formally and prove the equivalence.

For the proof of Theorem PTMT we only show that the product of two lower triangular matrices is again lower triangular. Provide a proof that the product of two upper triangular matrices is again upper triangular. Look to the proof of Theorem PTMT for guidance if you need a hint.