Normal but not self-adjoint

  • #1
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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^*?##
 

Answers and Replies

  • #2
DrDu
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I think every normal matrix can be written as A+iB where A and B are commuting hermitian matrices.
 
  • #3
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That's a really good idea, thanks a lot!!
 
  • #4
micromass
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Take ##n=1##, then every element of ##\mathbb{R}## is self-adjoint, while every element of ##\mathbb{C}## is normal.
 
  • #5
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Yes, that's a special case in ##M_1(\mathbb{C})!!##
 
  • #6
micromass
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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.
 

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