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.