I Normal but not self-adjoint

tommyxu3 · Apr 19, 2016

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 · Apr 19, 2016

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

tommyxu3 · Apr 19, 2016

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

micromass · Apr 19, 2016

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

tommyxu3 · Apr 21, 2016

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

micromass · Apr 21, 2016

tommyxu3 said: ↑

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.