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Homework Help: Exam question help

  1. Aug 8, 2008 #1
    two part question the first part i understand the second part i dont

    first part is find the eigenvectors and values of the matrix (4,-1 then below-4,4)

    giving me lamda=2 (1,2)T and lamda=6 (1,-2)T

    after this we have part 2 that i dont really understand how to do, ive done it before but have since forgot how to and cant find the corresponding notes

    B Use this result to find the sollution to the two dimensional linear map

    Xn+1= 4Xn-Yn

    Yn+1=-4Xn+4Yn

    with X0=1 and Y0=1

    thanks in advance for help, i also have a similar question but without the limits which to be honest i have never fully understood so i'll try to wrap my head arround this one then post that one after
     
  2. jcsd
  3. Aug 8, 2008 #2
    Since say, let A is a square matrix i.e. 2x2 matrix, then let

    [tex]x_{n+1} = 4x_{n} - y_{n} [/tex] (1)
    [tex]y_{n+1} = -4x_{n}+4y_{n} [/tex] (2)

    then this is a recursive function, notice that the coefficients are equal to your original matrix.
    ================================================
    Hint:
    Let A be an nxn matrix. Let [tex]r_{k+1} = Ar_{k} , k=(0,1,2....) [/tex]

    then to find a solution then take an eigenvector say [tex]r_{0}[/tex] and its eigenvalue [tex]\lambda [/tex] and let

    [tex]r_{k}= \lambda ^{k}x_{0}[/tex]

    Another hint: Let [tex]r_{0} = c_{1}v{1}+c_{2}v_{2}= [v_{1} v_{2}] \begin{bmatrix} c_{1} \\ c_{2} \end{bmatrix}[/tex]
    What are the constants c? What are the vectors v1, v2?
     
    Last edited: Aug 8, 2008
  4. Aug 8, 2008 #3
    sorry still totaly lost, could you go through it step by step so i can see whats going on?
     
  5. Aug 8, 2008 #4
    Couple of typos on my initial post. Reread it along with this post.

    Notice that your matrix is

    [tex]A= \begin{bmatrix} 4 & -1 \\4 & 4 \end{bmatrix}[/tex]

    and that you are given [tex]r_{0} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}[/tex]

    let [tex]v_{1}[/tex] be your first eigenvector, and [tex]v_{2}[/tex] be your second eigenvector.

    Notice that [tex]Av_{1}=\lambda_{1}v_{1}[/tex] and [tex]Av_{2} = \lambda_{2}v_{2}[/tex]

    Then

    [tex]\boxed{r_{k}=c_{1} \lambda_{1}^k v_{1} + c_{2} \lambda_{2}^k v_{2}}[/tex]

    Remember, notice that [tex]Ar_{k} = A(\lambda)^kr_{0} = \lambda^k(Ar_{0}) =\lambda^k (\lambda r_{0}) = \lambda^{k+1}r_{0} = r_{k+1}[/tex]
    Hope that helps
    ===============================
    Your first job is to find the [tex]c_{1},c_{2}[/tex] since you've stated that you've already found the lambdas. The eigenvectors, v_1 v_2, you've already found.
     
    Last edited: Aug 8, 2008
  6. Aug 8, 2008 #5

    HallsofIvy

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    Science Advisor

    Another way of looking at it is this: since your matrix, I'll call it A, has two independent eigenvectors, it can be "diagonalized". That is, there exist an invertible matrix U such that
    [tex]UAU^{-1}]= U\left[\begin{array}{cc}4 & -1 \\ -4 & 4\end{array}\right]U^{-1}= \left[\begin{array}{cc}6 & 0 \\ 0 & 2\end{array}\right]= D[/tex]
    and, of course, it follow from that that A= U-1DU. Now it should be easy to see that An= U-1DnU. For example, A2= (U-1DU)(U-1DU)= U-1D2U because the U-1 and U between the two "D"s cancel.

    In fact, you can take U-1 to be the matrix having the eignvectors as columns:
    [tex]\left[\begin{array}{cc}1 & 1 \\ -2 & 2\end{array}\right][/tex]
    so that U is its inverse
    [tex]\left[\begin{array}{cc}\frac{1}{2} & -\frac{1}{4} \\ \frac{1}{2} & \frac{1}{4}\end{array}\right][/tex]

    You should be able to check that
    [tex]\left[\begin{array}{cc}4 & -1 \\ -4 & 4\end{array}\right]=\left[\begin{array}{cc}1 & 1 \\ -2 & 2\end{array}\right]\left[\begin{array}{cc}6 & 0 \\ 0 & 2\end{array}\right]\left[\begin{array}{cc}\frac{1}{2} & -\frac{1}{4} \\ \frac{1}{2} & \frac{1}{4}\end{array}\right][/tex]
    So that
    [tex]\left[\begin{array}{cc}4 & -1 \\ -4 & 4\end{array}\right]^n=\left[\begin{array}{cc}1 & 1 \\ -2 & 2\end{array}\right]\left[\begin{array}{cc}6^n & 0 \\ 0 & 2^n\end{array}\right] \left[\begin{array}{cc}\frac{1}{2} & -\frac{1}{4} \\ \frac{1}{2} & \frac{1}{4}\end{array}\right][/tex]
    Mulitplying out the right side should give you the solution.
     
    Last edited by a moderator: Aug 8, 2008
  7. Aug 8, 2008 #6
    so the information that i currently have is lamda1=2 lamda2=6 V1=(1,2)T and V2=(1,-2)T

    What is r, K, and C and how do i go about finding them?
     
  8. Aug 8, 2008 #7
    so A^n=U^-1 . D^n . A . U

    where do my X0 and Y0 values fall into this i'm still confused but this is definatly closer to the way i remember doing this
     
  9. Aug 8, 2008 #8
    im going to walk home now and get some food but ill check this again in about a hour thanks guys
     
  10. Aug 8, 2008 #9

    HallsofIvy

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    Science Advisor

    ?? No one has said anything about "r, K, and C". konthelion did say that the solution can be written in the form
    [tex]{r_{k}=c_{1} \lambda_{1}^k v_{1} + c_{2} \lambda_{2}^k v_{2}}[/tex]
    where c1 and c2 are constants to be determined by the "initial conditions (X0= Y0= 1), the two [itex]\lambda[/itex]s are your eigenvalues, 2 and 6, and v1 and v2 are the corresponding eigenvalues.

    Since you got [itex]\lambda_1= 6[/itex] with [itex]v_1= \left[\begin{array}{c}1 \\ -2\end{array}\right][/itex] and [itex]\lambda_2= 2[/itex] with [itex]v_2= \left[\begin{array}{c} 1 \\ 2\end{array}\right][/itex], You have
    [tex]r_n= c_1 6^n \left[\begin{array}1 \\ -2\end{array}\right]+ c_2 2^n \left[\begin{array}{c}1 \\ 2\end{array}\right][/tex]
    Taking n= 0,
    [tex]r_n= c_1 \left[\begin{array}1 \\ -2\end{array}\right]+ c_2 \left[\begin{array}{c}1 \\ 2\end{array}\right]=\left[\begin{array}{c}c_1+ c_2 \\-2c_1+ 2c_2\end{array}\right]= \left[\begin{array}{c}1 \\ 1\end{array}\right][/tex]
    so we get the two equations c1+ c2= 1 and -2c1+ 2c2= 1. You can easily solve those to get c1= 1/4 and c2= 3/4. Then
    [tex]r_n= \frac{1}{4} 6^n \left[\begin{array}1 \\ -2\end{array}\right]+ \frac{3}{4} 2^n \left[\begin{array}{c}1 \\ 2\end{array}\right][/tex]
    [tex]= 6^n\left[\begin{array}{c}\frac{1}{4} \\-\frac{1}{2}\end{array}\right]+ 2^n\left[\begin{array}{c}\frac{3}{4} \\ \frac{3}{2}\end{array}\right][/tex]
    so that [itex]x_n= (1/4)6^n+ (3/4)2^n[/itex] and [itex]y_n= (-1/2)6^n+ (1/2)2^n[/itex].

    Doing the calculation I suggested gives you a matrix. Now multiply that matrix by the vector (or column matrix)
    [tex]\left[\begin{array}{c}x_0 \\ y_0 \end{array}\right]= \left[\begin{array}{c}1 \\ 1\end{array}\right][/tex]
    That should give you the same solution as above.
     
    Last edited by a moderator: Aug 8, 2008
  11. Aug 8, 2008 #10
    nice one i get it now thanks, having my tea right now but after that i'll go through this question and then have a look at the question i have without limits to see if i can do that and if not i'll put that one up as well, thanks for the help
     
  12. Aug 9, 2008 #11
    i had a question yesterday entitled exam question help which 2 people were kind enough to work through with me to a point were i now understand how to do this question, as i stated on there yesterday i have another question that is similar but this time it is a bit more complex, could someone please take a look at this one and if possible try to present it in the same way that hallsofivey did yesterday as this was the way that i understood the question.

    part (a) which i can do is find the solution to the two dimensional map

    Xn+1=Xn -Yn
    Yn+1=2Xn +4Yn

    with X0=1 and Y0=1

    giving me eivgenvalues of 2 and 3 with vectors of (1,-1)T and (1,-2)T respectivly

    After which i do the question just the same as the one that was answered for me above and i end up with hopefully the correct answer of Xn=-2 . 3n+3 . 2n

    and Yn=4.3n-3.2n

    hopefully that bits right. then i have part b of the question which i dont know how to do which is

    indicate how the second order map
    Xn+1=4xn-3xn-1

    can be expressed as a two dimensional map. by concidering the eigenvalues of the associated matrix show that Xn then theres a symbol that i have never seen before, it looks like a 8 on its side but part of the loop is missing on the right hand side then 3n for n large... thanks
     
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