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Homework Help: Matrices and eigenvalues. A comment in my answer.

  1. Dec 12, 2009 #1
    1. The problem statement, all variables and given/known data

    Hello and thanks again to anyone who has replied my posts. Your help is a great deal and really appreciated.

    I have the following homework question which I have answered and I want a comment if it is valid or illogical:

    We are given a matrix, with eigenvalues 3 and 7 respectively.

    We are asked to say if there other matrices with the same eigenvalues and if the set containing all these matrices is finite.



    2. Relevant equations
    the matrix given:

    2 1
    -5 8

    3. The attempt at a solution

    I have thought of the following answer: My point is that there are infinite matrix with the same eigenvalues that have different eigenvectors. So I say that the linear system:

    A(x1,y1)=λ1(x1,y1)
    A(x2,y2)=λ2(x2,y2)

    where A is a 2x2, has infinite solutions IF we take a11,a12,a21,a22(the elements of the matrix) as the variables of the linear system. The solutions will have x1,y1 and x2,y2 as constant parameters.

    Have i got something terribly wrong?
     
    Last edited: Dec 12, 2009
  2. jcsd
  3. Dec 13, 2009 #2

    HallsofIvy

    User Avatar
    Science Advisor

    No, that's perfectly correct.

    Another way to prove that is to note that the matrix
    [tex]\begin{bmatrix}3 & a \\ 0 & 7\end{bmatrix}[/tex]
    has eigenvalues 3 and 7 for any a.
     
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