1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Traceless hermitian matrices form groups?

  1. Nov 12, 2008 #1
    1. is the set of nxn traceless hermitian matrices under addition a group?
    2. is the set of nxn traceless hermitian matrices under multiplication a group?
    3. is the set of nxn traceless non-hermitian matrices under addition a group?


    question 1-I thought that traceless means trace=0 is this right? so what would the identity element be? it can't be the null matrix because it doesnt have an inverse, can anyone help? I haven't got around to the other questions but help is probably needed coz i dont like matrices
     
  2. jcsd
  3. Nov 12, 2008 #2
    I just realised in the first quesiton, the composition law is actually addition, so that makes the inverse of the identiy just putting a minus sign on all of its elements, which doesnt change the diagonal, which mean its still traceless, so it must be a group.

    for the second question closure isn't satisfied , the third one im not sure what to do...
     
  4. Nov 12, 2008 #3
    non-hermitian matrices don't include the identity.
     
  5. Nov 12, 2008 #4
    yes of course, thanks a lot
     
  6. Nov 12, 2008 #5

    Avodyne

    User Avatar
    Science Advisor

    Well, first of all, the identity element for addition is the matrix of all zeroes, not the identity matrix. Of course, this is also hermitian. But "non-hermitian" is often supposed to mean "not necessarily hermitian" rather than "definitely not hermitian". The answer depends on which meaning is implied.
     
  7. Nov 12, 2008 #6
    That was what I meant.

    "Not necessarily hermitian" just means all matrices. Then, there is no point in using such term.

    To me it seems safe to consider "non-hermitian" as "definitely not hermitian".
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Traceless hermitian matrices form groups?
  1. Hermitian Operators (Replies: 5)

  2. Hermitian Operators (Replies: 15)

Loading...