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Find the basis for both eigenvalues

  1. May 10, 2013 #1
    1. The problem statement, all variables and given/known data

    Given matrix A= {[39/25,48/25],[48/25,11/25]} find the basis for both eigenvalues.


    2. Relevant equations



    3. The attempt at a solution

    I row reduced the matrix and found both eigenvalues. I found λ = -1, and λ = 3. Then, I used diagonalization method [-1I2 - A 0] and [3I2 - A 0]. I got a basis of {[1,-3/4]} and {[1,4/3]}. However, I checked these but these were incorrect. I wasn't sure what I did wrong.
     
  2. jcsd
  3. May 10, 2013 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Row reduce the matrix? You can't find eigenvalues by row reduction!

    If [x, y] is an eigenvector corresponding eigenvalue -1, then
    [tex]\begin{bmatrix}\frac{39}{25} & \frac{48}{25} \\ \frac{49}{25} & \frac{11}{25}\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}-x \\ -y\end{bmatrix}[/tex]

    [tex]\begin{bmatrix}\frac{39}{25}x+ \frac{48}{25}y \\ \frac{48}{25}x+ \frac{11}{25}y\end{bmatrix}= \begin{bmatrix}-x \\ -y \end{bmatrix}[/tex]

    So we must have [itex]39x/25+ 48y/25= -x[/itex], which is the same as [tex]64x+ 48y= 0[/tex] and [tex]48x/25+ 11y/25= -y[/tex] which is the same as [tex]48x+ 36y= 0[/tex].

    Both of those equations reduce to a single equation of the form ax+ by= 0 so that y= (b/a)x.

    If [x, y] is an eigenvector corresponding eigenvalue 3, then
    [tex]\begin{bmatrix}\frac{39}{25} & \frac{48}{25} \\ \frac{49}{25} & \frac{11}{25}\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}3x \\ 3y\end{bmatrix}[/tex]

    [tex]\begin{bmatrix}\frac{39}{25}x+ \frac{48}{25}y \\ \frac{48}{25}x+ \frac{11}{25}y\end{bmatrix}= \begin{bmatrix}-x \\ -y \end{bmatrix}[/tex]

    So we must have [itex]39x/25+ 48y/25= 3x[/itex], which is the same as [tex]-36x+ 48y= 0[/tex] and [tex]48x/25+ 11y/25= 3y[/tex] which is the same as [tex]48x- 64y= 0[/tex]. Again, both of those equations reduce to a single equation.
     
  4. May 10, 2013 #3
    Okay, so given those two solutions, do I row reduce it to determine teh basis?
     
  5. May 11, 2013 #4

    Mark44

    Staff: Mentor

    No. The first equation in HallsOfIvy's post is 48x + 36y = 0, which is equivalent to 4x + 3y = 0. From that equation you can find the eigenvector that is associated with the eigenvalue λ = -1.

    The second equation is 48x - 64y = 0, or 3x - 4y = 0. Use this equation to find the eigenvector that is associated with the eigenvalue λ = 3.
     
  6. May 12, 2013 #5
    thank you
     
    Last edited: May 12, 2013
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