Normal matrices are the largest class of matrices which can be diagonalized by unitary transformations. They include Hermitian, anti‐Hermitian, unitary, real symmetric, and real orthogonal matrices, the primary types of interest to physicists. An aesthetically and practically satisfying synthesis as well as a significant saving of effort is achieved by studying normal matrices first and then deducing results for the various particular types subsequently. A simple constructive proof of the fundamental result that every normal matrix can be diagonalized by a unitary matrix is given.