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

Homework Help: Matrix General Expression

  1. Oct 3, 2006 #1
    Hi,

    I have a problem which involves determining a general expression in terms of k and n for the exponentiation of a matrix of the form:

    http://img242.imageshack.us/img242/2371/formwu2.png [Broken]

    The general expression I determined was accurate through a seperate process (which does not concern this post) was:

    http://img118.imageshack.us/img118/7174/genexpdf8.png [Broken]

    I am supposed to prove this expression is correct symbolically (ie I cannot use any examples with inputted k values) and I believe that the following should suffice:

    http://img156.imageshack.us/img156/1597/prooffg7.png [Broken]

    Does this seem correct? Also, is there a way that I can prove this without inputting values of n (ie entirely symbolical)?

    Any help is greatly appreciated.

    Matt
     
    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Oct 3, 2006 #2

    StatusX

    User Avatar
    Homework Helper

    Try induction.
     
  4. Oct 3, 2006 #3
    Unfortunately, I am in Grade 12 Discrete and we haven't done mathematical induction yet. I understand the basics of induction, but I have no idea how it should work with matrices and two variables. If anyone could help me with this, I would be grateful.
     
  5. Oct 3, 2006 #4

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    You would only being doing induction on the one variable "n". The set-up the the same as usual,

    base case: prove X^1 has that form (done)
    induction step: assuming your formula for X^n is correct, prove it is correct for X^(n+1). Just use X^(n+1)=X*(X^n) and use your formula for X^n.
     
  6. Oct 3, 2006 #5
    I believe this is what I am supposed to do:
    Proving the expression works for X^1, assuming that it is correct for X^n and then proving it is correct for X^(n+1):

    http://img301.imageshack.us/img301/6245/proof2yo4.png [Broken]

    However, there seems to be a problem.... the last 2 is to the exponent n+1, when it should be to the power of n, shouldn't it?
     
    Last edited by a moderator: May 2, 2017
  7. Oct 3, 2006 #6

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    There's some funny business going on there. To the right of your "X^(n+1)=..." it looks like you have X times your expression for X^(n+1), not X^n, since it has k^(n+1)'s all over and a 2^n in front.

    Then you had some problems when multiplying by this X two lines later. This may be some kind of "off by one" transcription error, but check it over carefully.
     
  8. Oct 3, 2006 #7
    I see the problem... I did it almost correctly on paper, and the only thing wrong with the equations I made was that I accidentally wrote in the n+1 (which I didnt do on paper :uhh: ) In any case, the real problem was that I DID substitute n+1 in for n in the exponent of the 2 when multiplying, so that error carried down.
    Thanks for all the help!
    p.s. Is there a way to prove it works for all negative integers too? or does this prove that?
     
  9. Oct 3, 2006 #8

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    This only proves it for positive n. For negative integers, just invert what you have. This only makes sense when X is invertable of course.
     
  10. Oct 3, 2006 #9
    So do it with n-1 instead of n+1?
     
  11. Oct 3, 2006 #10

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    I mean if n is positive, to find X^(-n) just take your formula for X^n and find the matrix inverse.
     
  12. Oct 3, 2006 #11
    I'm sorry but I don't understand - what do I do once I find the inverse? Find if it's true for n-1? And if so, assuming what?
     
  13. Oct 3, 2006 #12

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    Once you find the inverse you are done. Invert the formula for X^n and you get a formula for X^(-n). Since you've proven your formula for X^n was true for all positive n=1,2,3,4... this gives you a formula for all negative exponents n=-1,-2,-3... in one fell swoop.

    Ok, you're not quite done. It would be nice to show your formula for X^n works when n=0 as well, but that's simple enough to do. Now you will have a formula for X^n that's valid for all integers n.
     
  14. Oct 3, 2006 #13
    Easy enough.... Thank you shmoe, you have been most helpful :D!
     
  15. Oct 3, 2006 #14

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    Happy to help.

    By the way, good show on putting down all your work in a legible way:smile: . Since you are relatively new, you may not be aware that you can use https://www.physicsforums.com/showthread.php?t=8997" here, as in

    [tex]X=\left[\begin{array}{cc}k+1 & k-1\\k-1 & k+1\end{array}\right][/tex]

    (click on the image to see what you need to type to produce it).

    What you did was perfectly fine though! I only point it out as another option.
     
    Last edited by a moderator: Apr 22, 2017
  16. Oct 3, 2006 #15
    I did, but thanks for reminding me. This is part of a report for school, so I had the equations (well, most of them) previously made in that format anyway. But thanks for letting me know :smile:
     
    Last edited by a moderator: Apr 22, 2017
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook