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Matrix equation

  1. Jun 3, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the two numerical values of λ such that

    [itex]
    \left(\begin{array}{cc}4&3\\1&2\end{array}\right)
    \left(\begin{array}{cc}u\\1\end{array}\right)

    \left(\begin{array}{cc}u\\1\end{array}\right)
    [/itex]

    Hence or otherwise find the equations of the two lines through the origin which are invariant under the transformation of the plane defined by

    [itex]
    \left(\begin{array}{cc}x\prime\\y\prime\end{array}\right)
    =
    \left(\begin{array}{cc}4&3\\1&2\end{array}\right)

    \left(\begin{array}{cc}x\\y\end{array}\right)
    [/itex]

    2. Relevant equations


    3. The attempt at a solution
    I believe [itex]
    \left(\begin{array}{cc}4&3\\1&2\end{array}\right)
    [/itex] represents a non uniform scale and shear. If λ is a numerical value it can only scale uniformly which suggests there is no solution to the initial equation. However, my text book tells me the 2 numerical values are 5 and 1.

    I skipped this first part of the question and found the two invariant lines by setting [itex]y=mx+c [/itex] equal to [itex] y\prime=mx\prime+c[/itex]. If someone could help me understand the first part of the question I would be very appreciative.
     
  2. jcsd
  3. Jun 3, 2015 #2

    BvU

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    Homework Helper
    Gold Member

    Hi App,

    Can you write out the two equations that follow from the first matrix multiplication ? Eliminating ##\lambda## first seems the easiest to me; then the two values for u give the book values for ##\lambda##.
     
  4. Jun 3, 2015 #3

    Ray Vickson

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    The question is asking you to find the eigenvalues ##\lambda_1, \lambda_2## of the matrix
    [tex] A =\pmatrix{4&3\\1&2} [/tex]
    and to show that the eigenvectors of ##A## have the form
    [tex] \pmatrix{u\\1} [/tex]
    See, eg., http://tutorial.math.lamar.edu/Classes/DE/LA_Eigen.aspx .
     
  5. Jun 3, 2015 #4

    HallsofIvy

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    Staff Emeritus
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    The equation [itex]\begin{pmatrix}4 & 3 \\ 1 & 2 \end{pmatrix}\begin{pmatrix} u \\ 1 \end{pmatrix}= \lambda\begin{pmatrix} u \\ 1 \end{pmatrix} [/itex] is the same as the two equations [itex]4u+ 3= \lambda u[/itex] and [itex]u+ 2= \lambda[/itex]. That second equation is the same as [itex]u= \lambda- 2[/itex]. Replace u in the first equation with [itex]\lambda- 2[/itex] to get an equation in [itex]\lambda[/itex].
     
    Last edited: Jun 4, 2015
  6. Jun 4, 2015 #5
    Thanks for the help. I guess I wasn't taking into account the fact that u is not constant.

    I am still having difficulty understanding how the 2nd part of the question follows from the first. My book has not yet explicitly introduced me to eigenvectors, so I am presuming that the connection can be deduced from very rudimentary matrix principles.
     
  7. Jun 4, 2015 #6

    Ray Vickson

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    In fact, u is a constant---it is just the case that you don't know its value yet. Ditto for ##\lambda##.

    Basically, you just need to write down the two equations in ##u## that you get from rows 1 and 2 of your matrix; then you have two equations in the single unknown ##u##. (The equations have an undetermined parameter ##\lambda## in them.) In order for these two equations to be consistent, the value (or values) of ##\lambda## must be special, and figuring out these special values is the crux of your problem.
     
  8. Jun 4, 2015 #7

    Mark44

    Staff: Mentor

    Yes, it can, which should reinforce what an eigenvector is when it is formally defined.
     
  9. Jun 5, 2015 #8
    I think I get it now. So the value of λ is superfluous for the second part of the question, it is only really u that we are concerned with at this stage, since u defines the vectors which, when multiplied by λ define the 2 invariant lines through the origin.

    Thanks for all your help.
     
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