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Linear Algebra - Eigenvalue Problem

  1. Sep 16, 2012 #1
    1. The problem statement, all variables and given/known data

    Let there be 3 vectors that span a space: { |a>, |b>, |c> } and let n be a complex number.

    If the operator A has the properties:

    A|a> = n|b>
    A|b> = 3|a>
    A|c> = (4i+7)|c>

    What is A in terms of a square matrix?

    2. Relevant equations

    det(A-Iλ)=0

    3. The attempt at a solution

    I don't even know how to start. Can someone give me a starting hint?
     
  2. jcsd
  3. Sep 17, 2012 #2
    Imagine |a>=(1,0,0), |b>=(0,1,0), |c>=(0,0,1)
    What would the square matrix be then?
    Once you've got that, how can you relate your "hypothetical" square matrix to the correct one?
     
  4. Sep 17, 2012 #3

    HallsofIvy

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    Since A maps a three dimensional vector space to itself, it can be represented as a 3 by 3 matrix and so can be written
    [tex]\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}[/tex]

    Since you have basis vectors |a>, |b>, and |c>, they can be written as <1, 0, 0>, <0, 1, 0>, and <0, 0, 1>, respectively. What do you get when you multiply each of those by the matrix above?

    You are told that "A|a> = n|b>", that is the what you got by multiplying A by <1, 0, 0> must be equal to <0, n, 0> for this complex number n. Compare the two.

    Similarly, you are told that "A|b> = 3|a>". In other words, what you got by mutiplying A by <0, 1, 0> must be equal to <3, 0, 0>.

    Finally, you are told that "A|c> = (4i+7)|c>" so that what you got by multiplying A by <0, 0, 1> must be equal to <0, 0, 4i+7>.
     
  5. Sep 17, 2012 #4
    Ok, I didn't think of that. Writing them out in terms of specific numbers is helpful. However, why can we do that?

    I think I understand why they were written that way a little. Is it because they are orthogonal elements of a Hilbert Space and since they are orthogonal, we can define a coordinate system such that one is <1,0,0> and the other 2 are necessarily orthogonal to it and others, following the right hand rule?
     
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