I Normal but not self-adjoint

I think every normal matrix can be written as A+iB where A and B are commuting hermitian matrices.

 

Take ##n=1##, then every element of ##\mathbb{R}## is self-adjoint, while every element of ##\mathbb{C}## is normal.

 

And it can be generalized! Diagonal matrices with all real entries are self-adjoint, with complex entries are normal. Every normal operator can be diagonalized with unitary operators as transition matrices, so the general form of a self-adjoint matrix is ##UDU^*## with ##U## unitary and ##D## a diagonal matrix with real entries. The general form of a normal matrix is ##UDU^*## with ##U## unitary and ##D## a diagonal matrix with complex entries.