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!

Theoretical matrix problem

  1. Dec 18, 2013 #1
    We consider a matrix A, NxN.Show that if for every NxN matrix B we have AB=BA,then the matrix A is of the form:A=γI.
    When I first looked at this exercise,the first I thought was to assume that A=γI is true and replace A in AB=BA=>γB=Bγ=γB,which is true.But then I noticed that I have "=>" and not "<=>",so,since it just can't be that easy and simple,I guessed that's wrong.Any thoughts?
     
    Last edited: Dec 18, 2013
  2. jcsd
  3. Dec 18, 2013 #2
    Say the problem is stated If ##P \implies Q##.

    What you did was the converse: ##Q \implies P##, which isn't equivalent.

    The contrapositive IS equivalent however: ~##Q \implies## ~##P##. (Latex was messing up on me, sorry if you saw this before my edit)
     
    Last edited: Dec 18, 2013
  4. Dec 18, 2013 #3

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    scurty, if you want to put a twiddle inside of latex you can use \sim
     
  5. Dec 18, 2013 #4

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    That's trivial. The problem at hand is much harder. You need to show that the only way AB=BA is true for all NxN matrices B is if A is some scalar times the identity matrix.
     
  6. Dec 18, 2013 #5

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Or \neg. I edited scurty's post to use \neg just before you edited it to use \sim. Personally I like \neg better because the command says what you want, while \sim seems to be exactly the other way around.
     
  7. Dec 18, 2013 #6
    Yep,I figured out that was incorrect...So any tip on how to proceed?
     
  8. Dec 18, 2013 #7

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    When they say a property is true for ALL matrices B, the best way to approach the problem is to think of matrices B that give you a lot of information about what A is. If AB = BA and B is a specific matrix, you get a lot of information about how the entries of A are related. Ideally you would pick a matrix B which makes the information very simple, like specific entries being equal to zero, or two different entries being equal. Try playing around with some matrices B that aren't too hard to do calculations with.
     
  9. Dec 18, 2013 #8
    Thanks Office_Shredder and DH. I used Detexify but nothing was showing up for me and \~ wasn't working. Now I know for the future!
     
  10. Dec 19, 2013 #9
    I don't know how to proceed.I tried some B matrices to see what is going on with A.Then I assumed A=γI,so all fields are zero except for those on the main diagonal.What I got was that the elements of the diagonal were all the same,as expected.But I don't know how to generalize that and how the result to come out in a physical way.Should I consider random A,B matrices and assume that A=γI and to make zero all the non-main-diagonal elements of A in order to have all the main diagonal elements the same?But isn't that exactly the same with "=>"?Any further help?
     
  11. Dec 19, 2013 #10

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    You are looking at the problem the wrong way. Don't start with A=γI. Start with the given condition, that AB=BA for all matrices B. You have to prove that A=γI is the only solution that works for all matrices B.

    Hint: Look at the simplest of non-zero matrices B. For example, in the case of 2x2 matrices, look at the matrix ##B=\begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}##. What does this tell you about A? You need one more such simple matrix B to tell you that A must be a scalar times the 2x2 identity matrix.
     
  12. Dec 19, 2013 #11
    But the exercise says to show that A is of that form,not that A has ONLY that form.Why should I care if A is of any other form?And from 2x2 how can I generalize it to NxN?
     
  13. Dec 19, 2013 #12

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You still have it exactly backwards. You start with some NxN matrix about which almost nothing is know---no form, or anything is given to you. You are, however, told one bit of information, namely that AB = BA for all NxN matrices B. From that you are required to hammer down A to a special form, namely, a scalar times the identity matrix. Thefore, A = yI is not input, it is output.

    And, to answer your other question: you generalize it by sitting down and doing it! If you don't see how, then try the 3x3 case first. Don't agonize over it; just get started.
     
  14. Dec 19, 2013 #13

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    You are misreading the problem statement. It does not say to show that if the N×N matrix A is of the form A=γI then AB=BA for all N×N matrices B. It instead says to show that if AB=BA for all N×N matrices B then A is of the form A=γI. Another way to read "is of the form" is "must be of the form". Showing that AB=BA if A is a scalar times the identity does not show that A must be of that form because if P then Q is not the same as if Q then P.
     
  15. Dec 19, 2013 #14

    Mark44

    Staff: Mentor

    A bit off-topic, but I'm curious about a couple of things.

    "twiddle" = "tilde"?
    \neg I get, but what is \sim short for?
     
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: Theoretical matrix problem
  1. Matrix problem (Replies: 4)

  2. Matrix problem (Replies: 4)

  3. Matrix Problem (Replies: 6)

  4. Matrix problem (Replies: 11)

Loading...