[Math] Notion of a normal operator

functional-analysislinear algebrarepresentation-theory

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,

  1. $SO(3)$ is the group of rotations in $\mathbb R^3$
  2. A unitary matrix is one that represents an isometry
  3. 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)

Best Answer

The term "normal" here refers to orthogonality. Geometrically, a normal operator on $\mathbb C^n$ represents scaling by possibly different (complex) factors along different axes of an orthogonal coordinate frame. This, I think, is more intuitive than a Hermitian operator (which, of course, is also normal).

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