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Homework Help: Diagonalizable matrix

  1. May 4, 2006 #1
    [tex]M = \left(\begin{array}{cc}4&-1 \\ 2&7 \end{array}\right)[/tex]

    I need to show M^n as a formula of entries where n>0:

    so say [tex]M = \left(\begin{array}{cc}4&-1 \\ 2&7 \end{array}\right)[/tex] is diagonalizable: [tex]M = SDS^{-1}[/tex]

    then [tex]M^2=S^2 D^2 S^{-2} [/tex]

    and... [tex]M^3=S^3 D^3 S^{-3} [/tex]

    I can see from this that [tex]D^n = \left(\begin{array}{cc}\alpha^n&0 \\ 0&\beta^2 \end{array}\right)[/tex] assuming [tex]\alpha= \lambda_1[/tex] and that [tex]\beta= \lambda_2[/tex]

    [tex]\alpha= 5[/tex] and [tex]\beta= 6[/tex]

    [tex]D^n = \left(\begin{array}{cc}5^n&0 \\ 0&6^n \end{array}\right)[/tex]

    so to find S...

    [tex]\left(\begin{array}{cc}4-\alpha&-1 \\ 2&7-\beta \end{array}\right)[/tex]
    [tex]\left(\begin{array}{cc}-1&-1 \\ 2&2 \end{array}\right)[/tex]

    [tex]v_1= \left(\begin{array}{c}1\\ -1 \end{array}\right)[/tex]

    [tex]\left(\begin{array}{cc}2&1 \\ 2&1 \end{array}\right)[/tex]

    [tex]v_2= \left(\begin{array}{c}1\\ -2 \end{array}\right)[/tex]

    [tex]S = \left(\begin{array}{cc}1&1 \\-1&-2 \end{array}\right)[/tex]

    [tex]det(S)=-1[/tex]

    [tex]S^{-1} = \left(\begin{array}{cc}-1&-1 \\1&2 \end{array}\right)[/tex]

    this is where I am stuck, i dont know how to get S^n or S^-n

    any ideas?
     
    Last edited: May 4, 2006
  2. jcsd
  3. May 4, 2006 #2

    0rthodontist

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    Not true.
    M = SDS^-1.
    MM = SDS^-1SDS^-1
    How does this simplify?
     
  4. May 4, 2006 #3
    AKG helped me out and I realized that [tex]M^n=S D^n S^{-1} [/tex]

    what i dont understand now is now a diagonal matrix can be multiplied with other matricies to get a matrix with non-zero numbers?

    [tex]D^n = \left(\begin{array}{cc}5^n&0 \\ 0&6^n \end{array}\right)[/tex]

    so D multiplied by some other matrix will always be [tex] \left(\begin{array}{cc}a&0 \\ 0&b \end{array}\right)[/tex]

    so how does this produce the original matrix [tex]M = \left(\begin{array}{cc}4&-1 \\ 2&7 \end{array}\right)[/tex]?
     
  5. May 4, 2006 #4

    AKG

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    Multiply D with the matrix full of 1's. It has no zeroes. Anyways, we want to know how to find an invertible matrix S such that M = SDS-1. You were on the right track to finding S in that other thread.
     
  6. May 5, 2006 #5

    HallsofIvy

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    No, that's not true! Did you actually try it?

    How did you get D? Wasn't it D= SMS-1? So then M= S-1DS.

    You saw, of course, that the numbers on the diagonal of D are the eigenvalues of M. A standard way of finding S is to use the corresponding eigenvectors as columns of S.
     
  7. May 5, 2006 #6
    Lets give it a try:

    [tex]M = \left(\begin{array}{cc}4&-1 \\ 2&7 \end{array}\right)[/tex]

    [tex]M^n=S D^n S^{-1} [/tex]

    Find the eigenvalues:
    [tex]\left(\begin{array}{cc}4-\lambda&-1 \\ 2&7-\lambda \end{array}\right)[/tex]

    [tex]\lambda_1=5, \lambda_2=6[/tex]

    This mean:

    [tex]D = \left(\begin{array}{cc}5&0 \\ 0&6 \end{array}\right)[/tex]

    [tex]D^n = \left(\begin{array}{cc}5^n&0 \\ 0&6^n \end{array}\right)[/tex]

    To find the eigenvectors:
    [tex]\left(\begin{array}{cc}4-\lambda_1&-1 \\ 2&7-\lambda _1 \end{array}\right)[/tex]

    [tex]\left(\begin{array}{cc}4-5&-1 \\ 2&7-5 \end{array}\right)[/tex]

    [tex]\left(\begin{array}{cc}-1&-1 \\ 2&2 \end{array}\right)[/tex]

    Solving the matrix:
    [tex]\left(\begin{array}{ccc}-1&-1&0\\ 2&2 &0 \end{array}\right)[/tex]

    [tex]v_1=\left(\begin{array}{c}1\\-1 \end{array}\right)[/tex]

    [tex]\left(\begin{array}{cc}4-\lambda_2&-1 \\ 2&7-\lambda_2 \end{array}\right)[/tex]

    [tex]\left(\begin{array}{cc}4-6&-1 \\ 2&7-6 \end{array}\right)[/tex]

    [tex]\left(\begin{array}{cc}-2&-1 \\ 2&1 \end{array}\right)[/tex]

    Solving the matrix:
    [tex]\left(\begin{array}{ccc}-2&-1&0\\ 2&1 &0 \end{array}\right)[/tex]

    [tex]v_2=\left(\begin{array}{c}1\\-2 \end{array}\right)[/tex]

    The eigenspace is then:
    [tex]S=\left(\begin{array}{cc}1&1 \\ -1&-2 \end{array}\right)[/tex]

    [tex]det\left(\begin{array}{cc}1&1 \\ -1&-2 \end{array}\right)=-1[/tex]

    [tex]S^{-1}=\left(\begin{array}{cc}-1&-1 \\ 1&2 \end{array}\right)[/tex]

    So this means that:
    [tex]M= \left(\begin{array}{cc}1&1 \\ -1&-2 \end{array}\right)\left(\begin{array}{cc}5&0 \\ 0&6 \end{array}\right) \left(\begin{array}{cc}-1&-1 \\ 1&2 \end{array}\right) [/tex]

    [tex]M^n= \left(\begin{array}{cc}1&1 \\ -1&-2 \end{array}\right)\left(\begin{array}{cc}5^n&0 \\ 0&6^n \end{array}\right) \left(\begin{array}{cc}-1&-1 \\ 1&2 \end{array}\right) [/tex]

    When I solve this out.. I get:

    [tex]M^n= \left(\begin{array}{cc}-5^n+6^n&-5^n+2(6^n) \\ 5^n-2(6^n)&5^n-4(6^n) \end{array}\right) [/tex]

    When I sub in n=1, I do not get back the original matrix... could someone help?
     
    Last edited: May 5, 2006
  8. May 5, 2006 #7

    AKG

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    You don't solve a matrix, you solve an equation, and the equation you want to solve is:

    [tex]\left(\begin{array}{cc}-1&-1\\ 2&2 \end{array}\right)v_1 = \left (\begin{array}{c}0\\0\end{array}\right)[/tex]

    for v1, which coincidentally you have:
    Next
    Again, you're solving the equation:

    [tex]\left(\begin{array}{cc}-2&-1\\ 2&1 \end{array}\right)v_2 = \left(\begin{array}{c}0\\0\end{array}\right )[/tex]

    which you have already done.
    Just as a matrix is not something you solve, a matrix is not an eigenspace either. Moreover, it's not clear as to how you got this matrix. You're supposed to let S be the matrix whose rows (or columns, I can't remember) are the eigenvectors you found. You had vectors (1 -1)T and (1 -2)T, so I don't know where you're getting this S from.
    First of all, this matrix you're finding the determinat of is not the S you wrote down just a few lines above, and moreover it still isn't a matrix whose rows/columns are eigenvectors. It's yet another unexplained matrix. Thirdly, the determinant of that matrix there isn't -1, it's 0. Just look at the columns, they're the exact same, so the columns are clearly linearly dependent, so the determinant is 0. Of course, a very simply computation: (-1)(2) - (-1)(2) = 0 shows this as well. The original matrix for S that you had also has determinant 0, since its second row is just 2x the first, so they rows are dependant, hence determinant is 0. Again, a computation shows this: 1(-2) - (2)(-1) = 0.
    No matter which of the two matrices you're talking about, this is wrong since neither of those matrices are invertible (both have det 0). And I have no idea how you came up with this answer. I mean, if you did the thing you learned from high school: Take S, switch the things in positions a and d, replace c and d with their negatives, then divide the whole thing by det(S) [which you thought to be -1], you still wouldn't get what you have above, no matter which of the two matrices were taken to be S.
     
  9. May 5, 2006 #8

    AKG

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    Okay, it looks as though you've edited your post. Now everything's right up to the point where you find S-1 (well, you still don't solve matrices or call them eigenspaces, but anyways...). It looks like all you've done is taken S and divided by det(S) [which is indeed -1].
     
  10. May 5, 2006 #9
    sorry... still not so good with LaTeX.
     
  11. May 5, 2006 #10
    thanks for catching that mistake and then help AKG, I've got it figured out!
     
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