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Homework Help: Null space

  1. Sep 16, 2007 #1
    I need to find the null space of:

    [tex]\dotx \left(\begin{array}{cc}cos(\beta)-1&sin(\beta)e^{-i \alpha}\\sin(x)e^{i \alpha}&-cos(\beta)-1\end{array}\right)[/tex]

    [tex]\dotx \left(\begin{array}{cc}cos(\beta)-1&sin(\beta)e^{-i \alpha}\\sin(x)e^{i \alpha}&-cos(\beta)-1\end{array}\right) \binom{x}{y} = 0[/tex]

    I'm not sure how to go about doing this because I've been staring at:
    [tex](cos(\beta)-1)x=-sin(\beta)e^{-i \alpha} y[/tex]
    [tex]sin(\beta) e^{i \alpha} x = (-cos(\beta)-1) y[/tex]

    for a while now and I'm not sure how to get the x and y values
    Last edited: Sep 16, 2007
  2. jcsd
  3. Sep 16, 2007 #2


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    It's just a linear system of equations, isn't it? How do you normally solve systems of linear equations?
  4. Sep 16, 2007 #3
    by substitution, but the thing is I keep getting y=0 and x=0
  5. Sep 16, 2007 #4
    Does the obvious [tex]x = -sin(\beta) e^{-ia}, y = cos(\beta)-1[/tex] not work?
  6. Sep 16, 2007 #5
    sure, but i would like to know how to solve for that.
  7. Sep 16, 2007 #6
    Usually when you solve a problem like this-- i.e. a homogeneous linear system, you're going to end up with the trivial solution (x,y)=(0,0) and a surplus of other solutions with a free parameter.
    What you usually do is solve one of these equations and use a variable as a parameter.
    For example, if you solve the first equation you've been staring at for [tex]x[/tex], you get:
    [tex]x=-\frac{sin(\beta)e^{-i \alpha}}{cos(\beta)-1}y[/tex]
    And hence, you have [tex]\binom{-\frac{sin(\beta)e^{-i \alpha}}{cos(\beta)-1}y}{y}[/tex] as your null-space, y being a free parameter.
    You could have just as well have done it for x, with that as the free parameter.
    Ofcourse, you could always simplify this.
    You can do similarly for the second equation you got and obtain a basis for your null space.
  8. Sep 16, 2007 #7
    Notice that the two equations are not independent, so the solution, if one exists, would be a line. Such a line has to include the origin, so just eliminate/ignore one of the equations, and what you've got left should describe the line.
  9. Sep 16, 2007 #8
    thanks for the advice!

    In the grand scheme of things, I was trying to find the eigenvectors of:

    [tex]A=\dotx \left(\begin{array}{cc}cos(\beta)&sin(\beta)e^{-i \alpha}\\sin(x)e^{i \alpha}&-cos(\beta)\end{array}\right)[/tex]

    I got eigenvalues of +1 and -1

    if we took the -1 case, we have:

    [tex]\dotx \left(\begin{array}{cc}cos(\beta)+1&sin(\beta)e^{-i \alpha}\\sin(x)e^{i \alpha}&-cos(\beta)+1\end{array}\right) \binom{x}{y} = 0[/tex]

    my null space would be:

    [tex]x=\binom{-sin(\beta)}{(cos(\beta)+1)e^{i \alpha}}[/tex]

    However, this does not satisfy the eigenvalue equation: Ax=x

    [tex]\dotx \left(\begin{array}{cc}cos(\beta)-1&sin(\beta)e^{-i \alpha}\\sin(x)e^{i \alpha}&-cos(\beta)-1\end{array}\right) \binom{-sin(\beta)}{(cos(\beta)+1)e^{i \alpha}}=\binom{sin(\beta)}{(-cos(\beta)-1)e^{i \alpha}}[/tex]

    Not sure what this means
  10. Sep 16, 2007 #9
    It looks correct to me... The relevant equation is not Ax=x, but Ax = ax, where a is the eigenvalue.
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