The Complex Spectral Theorem for Hermitian Matrices

The real spectral theorem characterised orthonormal-eigenbasis existence by self-adjointness, on a real inner-product space. This note states the analogous result over $\mathbb{C}$ — and the characterising condition changes: it is normality ($TT^*=T^*T$, Definition 1 of the normal-operators note), the strictly weaker condition, not self-adjointness. This is not an inconsistency between the two theorems — it reflects a genuine structural difference between $\mathbb{R}$ and $\mathbb{C}$ that this note explains.

Learning objectives. By the end of this note you should be able to: