Results 1 to 5 of 5

Math Help - Steady States

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    225

    Steady States

    What are the limits as k -> infinity of the following:

    [.4 .2; .6 .8]^k[1; 0]

    [.4 .2; .6 .8]^k[0; 1]

    [.4 .2; .6 .8]^k

    I think the first two go to [1; 0] and the last goes to [0 0; 0 0], but am not sure.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Master Of Puppets
    pickslides's Avatar
    Joined
    Sep 2008
    From
    Melbourne
    Posts
    5,234
    Thanks
    27
    Quote Originally Posted by veronicak5678 View Post
    What are the limits as k -> infinity of the following:

    [.4 .2; .6 .8]^k
    [0.25 0.25;0.75 0.75]

    This should help you with the first 2.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2008
    Posts
    225
    Could you please explain how you got that answer?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Master Of Puppets
    pickslides's Avatar
    Joined
    Sep 2008
    From
    Melbourne
    Posts
    5,234
    Thanks
    27
    I used my calculator to find that matrix to a very high power. In this example either A^{50} or A^{100} will work.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    The more complete method of solution involves diagonalizing the matrix. Let's say that for a matrix A you could find an invertible matrix P and a diagonal matrix D such that A=PDP^{-1}. Then

    A^{2}=(PDP^{-1})(PDP^{-1})=PDDP^{-1}=PD^{2}P^{-1},

    A^{3}=(PDP^{-1})(PDP^{-1})(PDP^{-1})=PDDDP^{-1}=PD^{3}P^{-1}.

    More generally,

    A^{k}=PD^{k}P^{-1}.

    But the kth power of a diagonal matrix is just the (diagonal) matrix with the diagonal elements of the original diagonal matrix raised to the kth power.

    So you should be able to compute the limit exactly.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to formulize non-steady decay?
    Posted in the Pre-Calculus Forum
    Replies: 0
    Last Post: August 12th 2010, 11:05 PM
  2. Help with steady-state unemployment
    Posted in the Advanced Applied Math Forum
    Replies: 6
    Last Post: March 31st 2010, 03:13 PM
  3. steady state
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: March 20th 2010, 12:38 AM
  4. Steady State Vector
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: December 4th 2009, 05:47 PM
  5. Solving for homogenous steady states
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: November 10th 2009, 04:51 PM

Search Tags


/mathhelpforum @mathhelpforum