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Finding eigenvalues and eigenvectors of a matrix A

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

    Find the eigenvalues and eigenvectors of the matrix A = [2, 1; 8, 4]

    2. Relevant equations

    det(A - lambda I) = 0

    3. The attempt at a solution

    After expanding using the formula I have the equation (2 - [itex]\lambda[/itex])(4 - [itex]\lambda[/itex]) - 8

    Which gives [itex]\lambda[/itex] = 0, 6 (Should I be worried that I got a zero in my answer?)

    Then substituting the values for lambda back into the problem and using (A - [itex]\lambda[/itex]I)v = 0

    For [itex]\lambda[/itex] = 6
    [-4, 1; 8, -2] * [a; b] = [0; 0] and I get the eigenvector a[1; 4] from (-4a + b = 0)

    For [itex]\lambda[/itex] = 0
    [2, 1; 8, 4] * [a; b] = [0; 0] I get a second eigenvector of a[1; -2] from (2a + b = 0)

    Did I get the right eigenvalues and corresponding eigenvectors?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 1, 2011 #2

    Dick

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    Science Advisor
    Homework Helper

    Yes, you did. And it's easy for you to check as well. A*[1;4] should give you 6*[1;4] and A*[1;-2] should give you 0*[1;-2], right? Does it work?
     
  4. Aug 2, 2011 #3

    HallsofIvy

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    Staff Emeritus
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    Notice that the determinant of the matrix is 2(4)- 1(8)= 0. Since the determinant of a matrix is the product of its eigenvalues, 0 must be an eigenvalue.

     
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