Results 1 to 8 of 8
Like Tree4Thanks
  • 2 Post By emakarov
  • 1 Post By a tutor
  • 1 Post By emakarov

Math Help - Abstract Algebra Matrix Formulas

  1. #1
    Newbie
    Joined
    Sep 2012
    From
    United States
    Posts
    12

    Abstract Algebra Matrix Formulas

    Find a formula for the n-th power of any 2 x 2 upper-triangular real matrix,
    a...b
    0...d
    where n is a positive integer, and verify the formula by induction on n. Then, assuming a and d are nonzero, show the matrix is invertible and determine if there is a single formula that works for all integers n (i.e., when n < 0 too).

    So, first in order to derive a general formula for the matrix, I did out the multiplication. The second term in the first row is causing me problems. I find that the first term in the first row is a^n, and the last term in the last row is d^n, but the second term in the first row is causing problems. Below are my results for the second term for each n:
    n=1 : b
    n=2: ab + bd
    n=3: a^2b + abd + bd^2
    n=4: a^3b + a^2bd + abd^2 + bd^3

    I see a pattern, but am not sure how to represent the pattern in terms of n, so any help with this would be appreciated. I was thinking of something along the lines of a^(n-1)b + a^(n-2)bd... but I'm not really sure that's the way to go. Also, any other tips/advice while working on the rest of the problem would be awesome, and if there is a formula that works for negative integers of n as well.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Jan 2008
    From
    UK
    Posts
    484
    Thanks
    66

    Re: Abstract Algebra Matrix Formulas

    Take the b out and things look a little clearer.

    b
    b(a+d)
    b(a^2+ad+d^2)
    b(a^3+a^2d+ad^2+d^3)

    The expression in brackets can be written neatly as sum.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,545
    Thanks
    780

    Re: Abstract Algebra Matrix Formulas

    Also, \sum_{k=0}^na^{n-k}d^k=(a^{n+1}-d^{n+1})/(a-d).
    Thanks from a tutor and TheHowlingLung
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Sep 2012
    From
    United States
    Posts
    12

    Re: Abstract Algebra Matrix Formulas

    Quote Originally Posted by emakarov View Post
    Also, \sum_{k=0}^na^{n-k}d^k=(a^{n+1}-d^{n+1})/(a-d).
    I don't fully understand this. My experience with sum notation is brief. Could you explain this a little? When I look at that sum, I see it as an-0d0+an-1d1...+ an-ndn. Is that incorrect? Is that the correct way of representing the summation that's happening as n increases? I don't understand where you are getting (an+1-dn+1)/(a-d). Maybe I'm missing something?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Jan 2008
    From
    UK
    Posts
    484
    Thanks
    66

    Re: Abstract Algebra Matrix Formulas

    Example

    a^3-d^3=(a-d)(a^2+ad+d^2)

    \frac{a^3-d^3}{a-d}=a^2+ad~+d^2
    Thanks from TheHowlingLung
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,545
    Thanks
    780

    Re: Abstract Algebra Matrix Formulas

    Quote Originally Posted by TheHowlingLung View Post
    When I look at that sum, I see it as an-0d0+an-1d1...+ an-ndn.
    This is correct.

    Quote Originally Posted by TheHowlingLung View Post
    I don't understand where you are getting (an+1-dn+1)/(a-d).
    The easiest way to prove this is to multiply (a - d) by (and0+an-1d1...+ a0dn); most terms cancel out. See also here and here.
    Last edited by emakarov; September 6th 2012 at 03:12 AM.
    Thanks from TheHowlingLung
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Sep 2012
    From
    United States
    Posts
    12

    Re: Abstract Algebra Matrix Formulas

    Thanks guys! I appreciate the help.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,545
    Thanks
    780

    Re: Abstract Algebra Matrix Formulas

    Quote Originally Posted by emakarov View Post
    This is correct.

    The easiest way to prove this is to multiply (a - d) by (and0+an-1d1...+ a0dn); most terms cancel out. See also here and here.
    Edit: The first link does not work, but it's in the Math Help Boards forum. The part of the URL after the site name is correct.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: December 6th 2010, 03:03 PM
  2. Replies: 0
    Last Post: April 23rd 2010, 11:37 PM
  3. abstract algebra #2
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: January 28th 2010, 09:52 PM
  4. Abstract Algebra
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 17th 2008, 10:13 AM
  5. abstract algebra
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: February 27th 2008, 07:22 PM

Search Tags


/mathhelpforum @mathhelpforum