Normal operator intuition

Question

The formal definition for normal operator is $T$ $T^*$ = $T^*$ $T$. Can someone give me an intuition for what the definition means?

Cow we can connect the definition to this intuition?

Can you give also an intuition for why the eigenspaces are orthonormal to each other?

Why also for a given normal operator $T$, given $λ$ a eigenvalue why $(V_λ)^⊥$ is $T$-invariant?

I prefer a intuitive explanation, because I am familiar with the formal proof.

Thank you.

Answer 1

I'm not sure if you will call this "intuition" but there are two general facts worth knowing about adjoint operators and commuting operators even before talking specifically about normal operators:

1. If $W$ is a $T$-invariant subspace then $W^{\perp}$ is $T^{*}$-invariant. This follows from the fact that if $w \in W$ and $v \in W^{\perp}$ then $$ 0 = \left< Tw, v \right> = \left< w, T^{*} v \right>. $$
2. If $T,S$ are commuting operators (so $TS = ST$) then any eigenspace $V_{\lambda}$ of $T$ is $S$-invariant. Why? Because if $T(v) = \lambda v$ then $T(S(v)) = S(T(v)) = S(\lambda v) = \lambda S(v)$.

An operator $T$ is normal iff $T$ commutes with $T^{*}$. In particular, this means that:

1. The eigenspace $V_{\lambda}$ of $T$ is not only $T$-invariant but also $T^{*}$-invariant. But then $V_{\lambda}^{\perp}$ is $(T^{*})^{*} = T$-invariant. Hence we have a $T$-invariant decomposition $$ V = V_{\lambda} \oplus V_{\lambda}^{\perp}. $$ This is the "geometric" content behind the algebraic condition $T^{*}T = TT^{*}$ (at least over $\mathbb{C}$).
2. If $\mu \neq \lambda$ and $w$ is an eigenvector of $T$ associated to the eigenvalue $\mu$, we can write $w = v + v'$ with $v \in V_{\lambda}$ and $v' \in V_{\lambda}^{\perp}$ and then $$ \mu w = T(w) = T(v) + T(v') = \lambda v + T(v') = \mu v + \mu v'. $$ Hence we must have $\lambda v = \mu v$ and $T(v') = \mu v'$. Since $\mu \neq \lambda$ we must have $v = 0$ and so $w = v' \in V_{\lambda}^{\perp}$. Hence, different eigenspaces of $T$ are orthogonal to each other.

Answer 2

A Normal matrix is the most general complex matrix that has a complete orthonormal basis of eigenvectors. The definition was basically devised that way.

The normal matrix was a generalization of a Hermitian matrix, which has an orthonormal basis of eigenvectors with real eigenvalues. A normal matrix $N$ can be written as $N=A+iB$ where $A,B$ are commuting Hermitian matrices.