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System of Equations

  1. Nov 17, 2004 #1
    Hi! I'm trying to solve an equation system [tex]\vec{\pi}\mathbb{P} = \vec{\pi}[/tex] where [tex]\vec{\pi} = (\pi_1, \pi_2, \pi_3, \pi_4, \pi_5)[/tex] and [tex]\mathbb{P}[/tex] is a 5x5 matrix (constants). The problem is that the equation system is a bit to large to handle, at least for me. I remember that linear equation systems can be solved [tex]\mathbb{X}\vec{v} = \mathbb{Y} \Rightarrow \vec{v} = \mathbb{X}^{-1}\mathbb{Y}[/tex]. Is there anything similiar I can use to solve this system? Or can I solve it using maple or matlab?

    Thanks in advance,
    Nille
     
  2. jcsd
  3. Nov 17, 2004 #2
    hi,
    I am not sure whether i have correctly understood your problem.
    I think u will be able to solve the system of equation iteratively.
    consider,
    Ax=b
    where A is the constant 5 * 5 matrix, x is the variable vector(5 * 1) which you want to find out and b is again a constant vector(5 * 1).
    The above equation can be written as,

    (A+I - I)x = b, where I is the identity matrix.
    simplifying we get,
    x = (A+I)x -b.
    Hence x(k+1) = (A+I)x(k) - b, where k is the iteration number.
    One can start with some approximate value of the vector x at k=0.

    If you are trying to solve something like:
    Ax = x
    this is equivalent to finding the eigen vector corresponding to eigenvalue 1 for the matrix A.
    May be this might help you.
    In matlab there is a 'eig' command which gives you all the eigenvectors and eigen values of matrix.
     
    Last edited: Nov 17, 2004
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