I understand that a normal operator is an operator such that
$$ AA^\dagger = A^\dagger A $$ where $\dagger$ is the conjugate transpose.
However, what is the most intuitive way to "characterise" this? For example,
- $SO(3)$ is the group of rotations in $\mathbb R^3$
- A unitary matrix is one that represents an isometry
- A hermitian matrix is a generalization of symmetric, and is the "nicest" (diagonalizable, real eigenvalues, etc) of all matrices over $\mathbb C$
I was hoping for some sort of intuitive explanation of why I would care about normal matrices over $\mathbb C$ (maybe other reasons than the spectral theorem? Something more fundamental / geometric perhaps)