(a) A matrix is called normal if $A^{\dagger} A=A A^{\dagger}$. Let $A$ be a normal $n \times n$ complex matrix.

(i) Show that for any vector $\mathbf{x} \in \mathbb{C}^{n}$,

$$|A \mathbf{x}|=\left|A^{\dagger} \mathbf{x}\right|$$

(ii) Show that $A-\lambda I$ is also normal for any $\lambda \in \mathbb{C}$, where $I$ denotes the identity matrix.

(iii) Show that if $\mathbf{x}$ is an eigenvector of $A$ with respect to the eigenvalue $\lambda \in \mathbb{C}$, then $\mathbf{x}$ is also an eigenvector of $A^{\dagger}$, and determine the corresponding eigenvalue.

(iv) Show that if $\mathbf{x}_{\lambda}$ and $\mathbf{x}_{\mu}$ are eigenvectors of $A$ with respect to distinct eigenvalues $\lambda$ and $\mu$ respectively, then $\mathbf{x}_{\lambda}$ and $\mathbf{x}_{\mu}$ are orthogonal.

(v) Show that if $A$ has a basis of eigenvectors, then $A$ can be diagonalised using an orthonormal basis. Justify your answer.

[You may use standard results provided that they are clearly stated.]

(b) Show that any matrix $A$ satisfying $A^{\dagger}=A$ is normal, and deduce using results from (a) that its eigenvalues are real.

(c) Show that any matrix $A$ satisfying $A^{\dagger}=-A$ is normal, and deduce using results from (a) that its eigenvalues are purely imaginary.

(d) Show that any matrix $A$ satisfying $A^{\dagger}=A^{-1}$ is normal, and deduce using results from (a) that its eigenvalues have unit modulus.