Can Normal Matrices Be Non-Self-Adjoint?

[tommyxu3]

Hello everyone, I have a question. I'm not sure if it is trivial. Does anyone have ideas of finding a matrix ##A\in M_n(\mathbb{C})##, where ##A## is normal but not self-adjoint, that is, ##A^*A=AA^*## but ##A\neq A^*?##

 

[DrDu]

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

 

[tommyxu3]

That's a really good idea, thanks a lot!

 

[micromass]

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

 

[tommyxu3]

Yes, that's a special case in ##M_1(\mathbb{C})!##

 

[micromass]

Yes, that's a special case in ##M_1(\mathbb{C})!##

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.