Traceless hermitian matrices form groups?

Click For Summary

Homework Help Overview

The discussion revolves around the properties of nxn traceless hermitian matrices, specifically whether they form groups under addition and multiplication. The original poster raises questions about the identity element and inverses in the context of these matrices, expressing uncertainty about matrix concepts.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of traceless matrices and the implications for identity elements and inverses. There is a discussion about the nature of non-hermitian matrices and their inclusion of identity elements, with some participants questioning the terminology used.

Discussion Status

The conversation is ongoing, with some participants providing clarifications regarding the identity element for addition and the interpretation of non-hermitian matrices. Multiple interpretations of the terms are being explored, and there is no explicit consensus yet.

Contextual Notes

There is a noted confusion regarding the definitions of traceless and non-hermitian matrices, as well as the implications for group properties under different operations. The original poster expresses a lack of comfort with matrix concepts, which may influence the discussion.

TheIsingGuy
Messages
20
Reaction score
0
  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 doesn't have an inverse, can anyone help? I haven't got around to the other questions but help is probably needed coz i don't like matrices
 
Physics news on Phys.org
I just realized 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 doesn't 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 I am not sure what to do...
 
non-hermitian matrices don't include the identity.
 
weejee said:
non-hermitian matrices don't include the identity.

yes of course, thanks a lot
 
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.
 
Avodyne said:
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.

That was what I meant.

Avodyne said:
But "non-hermitian" is often supposed to mean "not necessarily hermitian" rather than "definitely not hermitian". The answer depends on which meaning is implied.

"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".
 

Similar threads

  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 1 ·
Replies
1
Views
2K
Replies
16
Views
3K
  • · Replies 9 ·
Replies
9
Views
2K
  • · Replies 4 ·
Replies
4
Views
3K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 7 ·
Replies
7
Views
8K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 7 ·
Replies
7
Views
3K
  • · Replies 2 ·
Replies
2
Views
3K