Results 1 to 4 of 4

Math Help - [SOLVED] Tricky little matrix problem

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    410

    [SOLVED] Tricky little matrix problem

    This exercise has been troubling me. It seems like it should be simple, but I can't seem to crack it.

    Let A=\left[\begin{array}{cc} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta}\end{array} \right]

    Find A^k, where k\in\mathbb{N}

    To solve this, I tried to find a pattern in the following:

    A^2=\left[\begin{array}{cc} \cos^2{\theta}-\sin^2{\theta} & 2\sin{\theta}\cos{\theta} \\ -2\sin{\theta}\cos{\theta} & \cos^2{\theta}-\sin^2{\theta}\end{array} \right]

    A^3=\left[\begin{array}{cc} \cos^3{\theta}-3\sin^2{\theta}\cos{\theta} & 3\sin{\theta}\cos^2{\theta}-\sin^3{\theta} \\ -3\sin{\theta}\cos^2{\theta}+\sin^3{\theta} & \cos^3{\theta}-3\sin^2{\theta}\cos{\theta}\end{array} \right]

    A^4=\left[\begin{array}{cc}\cos^4{\theta}-6\sin^2{\theta}\cos^2{\theta}+\sin^4{\theta}&4\sin  {\theta}\cos^3{\theta}-4\sin^3{\theta}\cos{\theta}\\-4\sin{\theta}\cos^3{\theta}+4\sin^3{\theta}\cos{\t  heta}&\cos^4{\theta}-6\sin^2{\theta}\cos^2{\theta}+\sin^4{\theta}\end{a  rray}\right]

    From this I observe that

    A^k=\frac{1}{k^2}\left[\begin{array}{cc}kf'(\theta)&k^2f(\theta)\\f''(\th  eta)&kf'(\theta)\end{array}\right]

    But I can't seem to generalize f(\theta) in terms of k. Any ideas?
    Last edited by hatsoff; September 1st 2008 at 03:50 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,965
    Thanks
    1785
    Awards
    1
    Did you notice that A^2  = \left[ {\begin{array}{cc}<br />
   {\cos (2x)} & {\sin (2x)}  \\<br />
   { - \sin (2x)} & {\cos (2x)}  \\<br />
\end{array}} \right]?
    I don't know if that helps?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by hatsoff View Post
    This exercise has been troubling me. It seems like it should be simple, but I can't seem to crack it.

    Let A=\left[\begin{array}{cc} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta}\end{array} \right]

    Find A^k, where k\in\mathbb{N}
    Think geometrically.
    This is a rotation matrix. Meaning if \bold{x}\in \mathbb{R}^2 then A\bold{x} represent rotation the vector \bold{x} counterclockwise by \theta (around the origin).

    Now A^2\bold{x} = A(A\bold{x}) is rotating \bold{x} twice by \theta i.e. 2\theta. And in general A^k is a rotation by k\theta. The matrix for the rotation is given by,
    A^k = \left[ \begin{array}{cc}\cos k\theta & \sin k\theta \\ - \sin k\theta & \cos k\theta \end{array} \right]
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by hatsoff View Post
    This exercise has been troubling me. It seems like it should be simple, but I can't seem to crack it.

    Let A=\left[\begin{array}{cc} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta}\end{array} \right]

    Find A^k, where k\in\mathbb{N}

    To solve this, I tried to find a pattern in the following:

    A^2=\left[\begin{array}{cc} \cos^2{\theta}-\sin^2{\theta} & 2\sin{\theta}\cos{\theta} \\ -2\sin{\theta}\cos{\theta} & \cos^2{\theta}-\sin^2{\theta}\end{array} \right]

    A^3=\left[\begin{array}{cc} \cos^3{\theta}-3\sin^2{\theta}\cos{\theta} & 3\sin{\theta}\cos^2{\theta}-\sin^3{\theta} \\ -3\sin{\theta}\cos^2{\theta}+\sin^3{\theta} & \cos^3{\theta}-3\sin^2{\theta}\cos{\theta}\end{array} \right]

    A^4=\left[\begin{array}{cc}\cos^4{\theta}-6\sin^2{\theta}\cos^2{\theta}+\sin^4{\theta}&4\sin  {\theta}\cos^3{\theta}-4\sin^3{\theta}\cos{\theta}\\-4\sin{\theta}\cos^3{\theta}+4\sin^3{\theta}\cos{\t  heta}&\cos^4{\theta}-6\sin^2{\theta}\cos^2{\theta}+\sin^4{\theta}\end{a  rray}\right]

    From this I observe that

    A^k=\frac{1}{k^2}\left[\begin{array}{cc}kf'(\theta)&k^2f(\theta)\\f''(\th  eta)&kf'(\theta)\end{array}\right]

    But I can't seem to generalize f(\theta) in terms of k. Any ideas?
    A thread of related interest: http://www.mathhelpforum.com/math-he...wer-100-a.html
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: March 29th 2010, 02:35 AM
  2. [SOLVED] Tricky Fraction Problem
    Posted in the Algebra Forum
    Replies: 2
    Last Post: February 12th 2009, 01:16 PM
  3. tricky matrix problem--please help!
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: December 17th 2008, 12:09 AM
  4. [SOLVED] Tricky volume problem
    Posted in the Calculus Forum
    Replies: 13
    Last Post: August 12th 2008, 03:20 PM
  5. [SOLVED] [SOLVED] Tricky Problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 2nd 2008, 11:23 PM

Search Tags


/mathhelpforum @mathhelpforum