Results 1 to 7 of 7

Math Help - Matrix Eigenvectors

  1. #1
    Newbie
    Joined
    May 2007
    Posts
    4

    Matrix Eigenvectors

    Hi just wondering if anybody could help me with a problem i've can't see to solve!

    I can work out the Eigenvalues fine

    Matrix A = [ 2 -1
    -1 1 ]

    Ax = Lx L = lamda

    | [A-LI] | = 0

    (2-L)(1-L) - 1 = 0

    L^2 - 3L + 1 = 0

    L(1) = 0.38
    L(2) = 2.62

    Now this is where i run into trouble. i try solve the simulataenous equations from this, these are.

    1. 1.62x1 - x2 = 0
    2. 0.62x2 - x1 = 0

    What do i do now, i've tried solving these but all i end up with is Cx1 = 0, which doesn't help anything, any help on what to do?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,705
    Thanks
    1637
    Awards
    1
    Quote Originally Posted by ct5845 View Post
    L(1) = 0.38
    L(2) = 2.62
    Now this is where i run into trouble. i try solve the simulataenous equations from this, these are.
    1. 1.62x1 - x2 = 0
    2. 0.62x2 - x1 = 0
    What exactly are you trying to do?
    Do you want to find the Eigenvectors?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2007
    Posts
    4
    yer the eigenvectors.

    The eigenvalues are correct, if someone could show me step by step how to work out one of the eigenvectors from one of the eigenvalues that would be a great help.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,705
    Thanks
    1637
    Awards
    1
    Here is an outline.
    Attached Thumbnails Attached Thumbnails Matrix Eigenvectors-eigen.gif  
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by ct5845 View Post
    Hi just wondering if anybody could help me with a problem i've can't see to solve!



    I can work out the Eigenvalues fine

    Matrix A = [ 2 -1
    -1 1 ]

    Ax = Lx L = lamda

    | [A-LI] | = 0

    (2-L)(1-L) - 1 = 0

    L^2 - 3L + 1 = 0

    L(1) = 0.38
    L(2) = 2.62

    Now this is where i run into trouble. i try solve the simulataenous equations from this, these are.

    1. 1.62x1 - x2 = 0
    2. 0.62x2 - x1 = 0

    What do i do now, i've tried solving these but all i end up with is Cx1 = 0, which doesn't help anything, any help on what to do?
    These are to the limits of numerical precision the same equation, so you realy
    only have one equation corresonding to lambda=0.38. Which is as it should
    be since the property of being an eigen vector tells you nothing about the
    magnitude of the vector.

    You have x2=1.62 x1, or the eigen vector is k[1, 1.62]'. Now you might want
    this to be a unit vector when you have something like:

    x ~= [0.5253, 0.8509]'

    Now try it with the other e-value.

    RonL
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    May 2007
    Posts
    4
    Hey, thanks thats answered half my questions but unfortunatly isn't right. ( is a past exam paper have the answer but didn't know how to get there )
    The unit vector part is the part i was missing thanks.

    So

    A = [ 2 -1, -1 1 ] Eigenvectors = [ -0.53, -0.85 ], [ -0.85,0.53 ]

    Eigenvalues = 0.38, 2.62

    So solving as simulataneous equations ( for L = 0.38 )
    (1) ( 2 - Lamda )x(1) + (-1)x(2) = 0 => 1.62x(1) - x(2) = 0
    (2) (-1)x(1) + (1-Lamda)x(1) = 0 => -x(1) + 0.62x(2) = 0

    now obviously from this we get the answer in the above post

    1.62x(1) = x(2)
    and 0.62x(2) = x(1)

    So getting the unit vector = [0.53, 0.85], any idea why the signs would be different to the 'answer', I personally can't see why unless for some reason you take 1.62x(1) from a side in (1) so

    (1) -x(2) = -1.62x(1)

    but i'm not sure why on earth you would do this? Again thanks for any help given.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by ct5845 View Post
    Hey, thanks thats answered half my questions but unfortunatly isn't right. ( is a past exam paper have the answer but didn't know how to get there )
    The unit vector part is the part i was missing thanks.

    So

    A = [ 2 -1, -1 1 ] Eigenvectors = [ -0.53, -0.85 ], [ -0.85,0.53 ]

    Eigenvalues = 0.38, 2.62

    So solving as simulataneous equations ( for L = 0.38 )
    (1) ( 2 - Lamda )x(1) + (-1)x(2) = 0 => 1.62x(1) - x(2) = 0
    (2) (-1)x(1) + (1-Lamda)x(1) = 0 => -x(1) + 0.62x(2) = 0

    now obviously from this we get the answer in the above post

    1.62x(1) = x(2)
    and 0.62x(2) = x(1)

    So getting the unit vector = [0.53, 0.85], any idea why the signs would be different to the 'answer', I personally can't see why unless for some reason you take 1.62x(1) from a side in (1) so

    (1) -x(2) = -1.62x(1)

    but i'm not sure why on earth you would do this? Again thanks for any help given.

    If [0.53, 0.85]' is a unit e-vector then so is [-0.53, -0.85]' and vice versa.
    They are both unit e-vectors for the given matrix correspomding to the same
    e-value.

    RonL
    Last edited by CaptainBlack; May 17th 2007 at 03:32 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Eigenvectors of this 3x3 matrix
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: May 18th 2010, 02:27 PM
  2. Matrix Eigenvectors
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 26th 2010, 01:07 AM
  3. eigenvectors and orthogonal matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 7th 2010, 10:14 AM
  4. Matrix diagonalization & eigenvectors
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: October 12th 2009, 06:58 AM
  5. Eigenvectors and eigenvalues of a matrix
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: May 6th 2008, 08:29 AM

Search Tags


/mathhelpforum @mathhelpforum