1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quick question on eigenvalues

  1. Mar 16, 2008 #1
    Let's say that I have to construct a 2 X 2 matrix from a second-order differential equation, turning it into a system of first order linear equations, and find its eigenvalues. I'll have two variables that correspond to the two columns in the matrix.

    If I swap columns, I end up with two different characteristic polynomials, and thus different sets of eigenvalues (this is the problem). How do I know which variable should correspond to the first column, and which variable should correspond to the second column?
     
  2. jcsd
  3. Mar 17, 2008 #2
    If you switch the columns, you switch the order of the variables.
     
  4. Mar 17, 2008 #3

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You spent a course learning linear arithmetic, so use it. :wink:

    Swapping columns is a matrix operation: if you started with the equation

    [tex]A x = y[/tex]

    and you want to replace A with AC, where C is the column swap elementary matrix... i.e.

    [tex]
    C = \left(
    \begin{array}{cc}
    0 & 1 \\
    1 & 0
    \end{array}
    \right)
    [/tex]

    Can you think of any way to modify the equation Ax=y (without changing its validity!) to make a C appear somewhere in it? (Preferably just to the right of the A)
     
  5. Mar 17, 2008 #4
    Thanks for the replies :)

    My math professor helped me out this morning, and I found out that if I have a second order differential equation corresponding to an equation y(t), and I let dy/dt = v, then the "rule of thumb" is to let the coefficient matrix A have its first column correspond to the variable y and let the second column correspond to the derivative of y(t), v (this way I know which coefficients in A to subtract [tex]\lambda[/tex] from). But now I want to know what you're talking about! So I'll take a shot at your question :)

    I believe your question is, how can I place the column swap matrix C into the equation Ax = y, without the equation losing its validity. To do this, I believe I would multiply both sides of Ax = y by C, such that C appears between A and x, and to the left of y. So I would have ACx= Cy. I worked this out on paper and I see that this does indeed swap the columns in A and the rows in y, which wouldn't change my answer, so it works out. Please let me know if I'm mistaken.
     
  6. Mar 17, 2008 #5

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    That doesn't quite work -- remember that the operations you have available are "left multiply" and "right multiply"... so you can't just insert it anywhere you want.

    The method I was hinting at is this idea: If I want to right-multiply A by C, I can undo that by right-multiplying by C-inverse. In this case, the inverse of C is itself, so I have the following derivation:

    Ax = y
    A (CC) x = y
    (AC) (Cx) = y

    The procedure you used shouldn't work in general: if Ax=y, then usually ACx=Cy will be false. In fact, that implies:

    ACx = Cy
    ACx = C(Ax)
    ACx = CAx
    (AC - CA) x = 0

    so it will only work when the solution for x happens to be a nullvector of (AC - CA).
     
  7. Mar 17, 2008 #6
    I see, so this works out because C*C = I, where I would be the 2 X 2 identity matrix, and AI = A.
     
  8. Mar 17, 2008 #7

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Right. And if you did a more complicated column operation to A, you can hopefully work out what happens to x to neutralize it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Quick question on eigenvalues
  1. Eigenvalues question. (Replies: 5)

  2. Quick Question. (Replies: 4)

  3. Quick Question (Replies: 13)

  4. Eigenvalue questions (Replies: 2)

Loading...