Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Normal but not self-adjoint

  1. Apr 19, 2016 #1
    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^*?##
     
  2. jcsd
  3. Apr 19, 2016 #2

    DrDu

    User Avatar
    Science Advisor

    I think every normal matrix can be written as A+iB where A and B are commuting hermitian matrices.
     
  4. Apr 19, 2016 #3
    That's a really good idea, thanks a lot!!
     
  5. Apr 19, 2016 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Take ##n=1##, then every element of ##\mathbb{R}## is self-adjoint, while every element of ##\mathbb{C}## is normal.
     
  6. Apr 21, 2016 #5
    Yes, that's a special case in ##M_1(\mathbb{C})!!##
     
  7. Apr 21, 2016 #6

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Normal but not self-adjoint
  1. Making T self adjoint (Replies: 8)

Loading...