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Linear Algebra Eigenvector Properties

  1. Mar 14, 2013 #1
    1. The problem statement, all variables and given/known data

    True/False: If true give a proof, if false give a counterexample.
    a)
    If A and B have the same eigenvector X, then A+B should also have the same eigenvector, X.
    b)
    if A has an eigenvalue of 2, and B has an eigenvalue of 5, then 7 is an eigenvalue of A+B




    2. Relevant equations



    3. The attempt at a solution

    for b):
    2 0 2 3
    0 2 has eigenvalue of 2; 3 2 has eigenvalue of 5

    When I add them together (A+B) you get 4 3
    3 4

    Then I found an eigenvalue of 7; Is this correct?
    Or the property of A+B != eigenvalueA + eigenvalueB is always correct? But this question's wording is kind of weird, because it said if its true give a counterexample ...


    for a) I think it is false,...not entirely sure though.
     
  2. jcsd
  3. Mar 14, 2013 #2

    epenguin

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    Homework Helper
    Gold Member

    That's not what I read.
     
  4. Mar 14, 2013 #3
    Ok I am stupid, I read the question wrong so I confused myself...here goes my mark...
     
  5. Mar 15, 2013 #4

    HallsofIvy

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    Staff Emeritus
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    (a) asks you to show that If X is an eigenvector for both A and B then it is an eigenvector for A+ B. If X is an eigevector of A, then [itex]AX= \lambda_A X[/itex] for some number [itex]\lambda_A[/itex]. If X is an eigenvector of B, then [itex]BX= \lambda_B X[/itex] for some number [itex]\lambda_B[/itex]. Now, what can you say about (A+ B)X?
     
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