Normal Matrices

Why

Which matrices have orthogonal eigenvectors?

Definition

A normal matrix is a matrix which has orthogonal eigenvectors. It commutes with its (conjugate) transpose.

Notation

If $A \in \C ^{d \times d}$ is normal then there exists an orthonormal matrix $Q \in \C ^{d \times d}$ and a diagonal matrix $\Lambda \in \C ^{d \times d}$ so that $A = Q\Lambda Q^\top $.