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Solve this eigen value problem

  1. Oct 2, 2008 #1
    plz solve this eigen value problem

    A nd B are matrices of order n*n.now it is given that sum of each row of A is 2 nd that of B is 1...then show that 2 is an eige value of the product matrix AB
     
  2. jcsd
  3. Oct 4, 2008 #2

    statdad

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    Re: plz solve this eigen value problem

    Let [tex] \vec v [/tex] be an [tex] n \times 1 [/tex] vector like this:

    [tex]
    \vec v' = \left[\frac 1 n \frac 1 n \dots \frac 1 n \right]
    [/tex]

    Then compute all of

    [tex]
    \begin{align*}
    & A \vec v \\
    & B \vec v\\
    & (AB) \vec v
    \end{align*}
    [/tex]

    and remember that for any matrix [tex] W [/tex] and vector [tex] \vec z [/tex], if there is a scalar [tex] k [/tex] such that

    [tex]
    W \vec z = k \vec z
    [/tex]

    then [tex] k [/tex] is an eigenvalue of the matrix [tex] W [/tex].
     
    Last edited: Oct 4, 2008
  4. Oct 7, 2008 #3
    Re: plz solve this eigen value problem

    how to compute (AB)v
     
  5. Oct 7, 2008 #4

    statdad

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    Re: plz solve this eigen value problem

    Here is a small example (note: the rows in this matrix do not sum to either 1 or 2, as doing that would be solving the problem for you. However, precisely the same steps work)

    [tex]
    \begin{align*}
    A &= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\\
    \vec v & = \begin{bmatrix} \frac 1 2 & \frac 1 2 \end{bmatrix}'
    \end{align*}
    [/tex]

    Then
    [tex]
    A \vec v = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 1/2 \\ 1/2 \end{bmatrix} = \begin{bmatrix} {(1+2)}/2 \\ {(3+4)}/2 \end{bmatrix}
    [/tex]

    so in this case, and in every case, the product [tex] A \vec v [/tex] has as its entries the means of the rows of [tex] A [/tex].
     
  6. Oct 7, 2008 #5
    Re: plz solve this eigen value problem

    but why should i try to get mean here????i mean i can take v (transpose) as[1 1 1 .... 1](n times)
    why r u taking [1/n 1/n ......... 1/n]?????
     
  7. Oct 7, 2008 #6

    Dick

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    Re: plz solve this eigen value problem

    Go ahead. Just use [1,1,1...]. (Not that there's anything wrong with using [1/n,1/n,...], you'll get the same result in the end).
     
  8. Oct 7, 2008 #7

    statdad

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    Re: plz solve this eigen value problem

    Just try the multiplication (or use Dick's suggestion) and notice how the result compares to the vector [tex] \vec v [/tex].

    Remember that if [tex] A \vec v = k \vec b [/tex] then [tex] k [/tex] is an eigenvalue of the matrix [tex] A [/tex].
     
  9. Oct 7, 2008 #8
    Re: plz solve this eigen value problem

    so the solution is using v=[1 1 1 1 ... 1]
    we can easily get Av=2v & Bv=1v
    then(AB)v=A(Bv)=A(1v)=Av=2v
    so 2 is an eigen value of AB................
    is this right solution?????????????????????
     
  10. Oct 7, 2008 #9

    Dick

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    Re: plz solve this eigen value problem

    (AB)v=2v. That sure looks like it says 2 is an eigenvalue to me.
     
  11. Oct 7, 2008 #10
    Re: plz solve this eigen value problem

    is that solution correct man???????????
     
  12. Oct 7, 2008 #11

    Dick

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    Re: plz solve this eigen value problem

    Do you have any doubts??????? You don't need me to approve your solution. If you believe in it go for it.
     
  13. Oct 10, 2008 #12

    HallsofIvy

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    Re: plz solve this eigen value problem

    Surely you meant [tex]A\vec v= k \vec v [/tex]?
    Which is the definition of "eigenvalue".
     
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