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Proving eigenvalues?

  1. Aug 20, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove: If a, b, c, and d are integers such that a+b=c+d, then
    A=
    [a b]​
    [c d] ​
    has integer eigenvalues, namely,[itex]λ_1{}[/itex]=a+b and [itex]λ_2{}[/itex]=a-c

    2. Relevant equations

    No relevant equation.

    3. The attempt at a solution

    No idea :(
     
  2. jcsd
  3. Aug 20, 2011 #2

    fzero

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    Do you know how to compute the eigenvalues of a 2x2 matrix? Try that first before applying the additional information given in the problem.
     
  4. Aug 20, 2011 #3

    HallsofIvy

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    You don't really need to calculate the eigenvalues, you are only asked to show that a+b and a- c are eigenvalues- and to do that you use the definition of "eigenvalue".

    That is, do there exist values, x and y, such that
    [tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= (a+b)\begin{bmatrix}u \\ v\end{bmatrix}[/tex]
    or values u and y such that
    [tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}u \\ v\end{bmatrix}= (a-b)\begin{bmatrix}u \\ v\end{bmatrix}[/tex]
    ?
     
  5. Aug 20, 2011 #4

    I like Serena

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    Welcome to PF, hadizainud! :smile:

    The methods of fzero and HallsofIvy will bring you your answer.
    Just for fun, here's yet another method.

    The product of the eigenvalues is equal to the determinant.
    The sum of the eigenvalues is equal to the trace.
    In a quadratic equation the roots are uniquely identified by their product and their sum.
    Use a+b=c+d to eliminate d in your equations.
     
  6. Aug 20, 2011 #5

    micromass

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    Hmm, that's cute :smile:
     
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