Notes - NLA MT25, Structured matrices

@Define an upper Hessenberg matrix $A \in \mathbb C^{m \times n}$.

$A _ {ij} = 0$ if $i > j + 1$, i.e. all entries below the subdiagonal are zero.

@State a result about a characterisation of real symmetric matrices in terms of diagonalisability.

A real matrix $A \in \mathbb R^{n \times n}$ is symmetric iff it has real eigenvalues and is orthogonally diagonalisable.

@Prove that a real matrix $A \in \mathbb R^{n \times n}$ is symmetric iff it has real eigenvalues and is orthogonally diagonalisable.

@todo once covered in lectures.

@State a result about a characterisation of normal matrices in terms of diagonalisability.

A matrix $A \in \mathbb C^{n \times n}$ is normal (i.e. $A^\ast A = A A^\ast$) iff it is unitarily diagonalisable.

@Prove that a matrix $A \in \mathbb C^{n \times n}$ is normal (i.e. $A^\ast A = A A^\ast$) iff it is unitarily diagonaliable.

@todo, once covered in lectures.