Results 1 to 10 of 10

Thread: Equation for eigenline

  1. #1
    Member
    Joined
    Jan 2009
    Posts
    142

    Equation for eigenline

    Hi

    Im having trouble with finding the equation of an eigenline. I can see that

    $\displaystyle \left(\begin{array}{cc} 1 & 2\\
    3 & 2\end{array}\right)\left(\begin{array}{c}
    x\\
    y\end{array}\right)=4\left(\begin{array}{c}
    x\\
    y\end{array}\right)
    $

    gives the following equations

    $\displaystyle x+2y=4x$
    $\displaystyle 3x+2y=4y$

    Which is

    $\displaystyle -3x+2y=0$
    $\displaystyle 3x-2y=0$

    But I'm not sure how to reduce these equations to find the equation of the eigenline

    $\displaystyle 3x-2y=0$

    It's probably simple but I just can't see it.

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    You simply then constuct a basis vector that fits that equation: $\displaystyle (x, x3/2)$. For example, $\displaystyle (1, 3/2)$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jan 2009
    Posts
    142
    Hi

    Thanks for the reply but I must be being dense, could you elaborate please.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    The work you have done has given you conditions for the $\displaystyle x$ and $\displaystyle y$. So a vector is an eigenvector if and only if it it obeys those restrictions and is non-zero. That is to say, it is of the form $\displaystyle (x, x3/2)$ where $\displaystyle x \in F \setminus \{0\}$. Clearly, $\displaystyle (1, 3/2)$ fulfills those conditions.

    The bit which says "v an eigenvector if and only if it it obeys those restrictions" holds because:
    $\displaystyle \Rightarrow$ as $\displaystyle (x,y)$ was an arbitrary eigenvector (for the eigenvalue 4)
    $\displaystyle \Leftarrow$ any vector that adheres to those conditions must also be an eigenvector for that eigenvalue as you just showed that it fulfils the conditions to be an eigenvector ($\displaystyle v \lambda = M v$).

    I apologise if that still doesn't make sense! Feel free to ask again. I still haven't got to grips with Linear algebra, so I can totally understand what you mean about feeling dense!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Jan 2009
    Posts
    142
    Hi

    Thanks, I've got it.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,840
    Thanks
    3063
    You do understand, don't you, that the y= (3/2)X that you got from swlabr is the same as 2y= 3x or 3x-2y= 0, exactly the equation, you posted originally?

    The "eigenline" is given by 3x- 2y= 0!
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Jan 2009
    Posts
    142
    Hi

    Yes I do. The problem was that for one stupid moment I could'nt see how the eigen line equation was found.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Newbie
    Joined
    Jan 2013
    From
    Margate
    Posts
    2

    Re: Equation for eigenline

    Hi, I have come across this same example, and I am the same I don't get it. The text says "these equations both reduce to 3x-2y=0. Thus the eigenlines corresponding to the eigenvalue k =4 has the equation y=(3/2)x". I understand that 3x-2y=0 is the same as y=(3/2)x but cannot seem to get my head around how the equations reduce to 3x-2y=0. I'm sure this is just basic manipulation but I can't see it.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,840
    Thanks
    3063

    Re: Equation for eigenline

    Are you referring to the equations x+ 2y= 4x and 3x+ 2y= 4y?

    Subtract 4x from both sides of x+ 2y= 4x to get -3x+ 2y= 0. Subtract 4y from both sides of 3x+ 2y= 4y to get 3x- 2y= 0. Those are clearly the same (one is the other multiplied by -1).
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie
    Joined
    Jan 2013
    From
    Margate
    Posts
    2

    Re: Equation for eigenline

    Thanks for that, I have tried this on an assignment question, could someone please check? I'm not after the answer, just where I'm going wrong.
    A=5 7
    -2 -4
    I believe the eigen values to be k=3 and -2

    This gives me the eigen equations for k=3 5x+7y = 3x and -2x-4y =3y

    These I believe cancel down to y=(2/7)x

    Which then gives me an eigen vector of 7
    2

    I have done similar with the second eigen value, I know these are wrong because a subsequent part of the question asks for the pdp-1 which I have carried out and does no bring me back to the original matrix.

    I believe the error is with the cancel down which is the bit I cannot grasp...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Apr 11th 2011, 01:17 AM
  2. Partial differential equation-wave equation - dimensional analysis
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: Aug 28th 2009, 11:39 AM
  3. Replies: 2
    Last Post: May 18th 2009, 12:51 PM
  4. Replies: 2
    Last Post: Apr 28th 2009, 06:42 AM
  5. Replies: 1
    Last Post: Oct 23rd 2008, 03:39 AM

Search Tags


/mathhelpforum @mathhelpforum