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Eigenvectors and eigenvalues

  1. Dec 30, 2017 #1
    1. The problem statement, all variables and given/known data
    Find the eigenvalues and eigenvectors of the matrix
    ##A=\matrix{{2, 0, -1}\\{0, 2, -1}\\{-1, -1, 3} }##

    What are the eigenvalues and eigenvectors of the matrix B = exp(3A) + 5I, where I is

    the identity matrix?


    2. Relevant equations


    3. The attempt at a solution
    So i've found the eigenvectors for A to be ##\frac{1}{\sqrt{6}}\vec{1,1,-2}##, ##\frac{1}{\sqrt{3}}\vec{1,1,1}##, ##\frac{1}{\sqrt{2}}\vec{-1,1,0}## with eigenvalues 4,1 one 2 respectively. but i don't know how to do the second part

    Many thanks
     
  2. jcsd
  3. Dec 30, 2017 #2

    PeroK

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    Can you calculate the eigenvalues of ##exp(A)##?
     
  4. Dec 30, 2017 #3
    They are the exponentials of the eigenvalues of A
     
  5. Dec 30, 2017 #4

    PeroK

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    Well, that's a good start. What about ##exp(3A)##?
     
  6. Dec 30, 2017 #5
    ##e^{3\lambda}##?
     
  7. Dec 30, 2017 #6

    PeroK

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    How could you prove that if you are not sure? Hint: it's not hard. Try letting ##B = 3A##
     
  8. Dec 30, 2017 #7
    the eigenvalues of ##exp(B)## are ##e^b## but ##b=3a## where a are the eigenvalues of A for ##B=3A##. Hence the eigenvalues are ##e^{3a}##
     
  9. Dec 30, 2017 #8

    PeroK

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    Yes. Although, I would start with something like:

    Let ##v## be an eigenvector of ##A## with eigenvalue ##\lambda \dots##
     
  10. Dec 30, 2017 #9
    Okay. But how do you find the eigenvalues of ##exp(3A)+5I##?
     
  11. Dec 30, 2017 #10

    PeroK

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    I thought you had worked it out. Where do you think you are stuck?
     
  12. Dec 30, 2017 #11

    PeroK

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    ##\dots Bv = (\exp(3A) + 5I)v = \dots##

    Does that help?
     
  13. Dec 30, 2017 #12

    Ray Vickson

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    For future reference: you can format a matrix nicely as
    $$A = \pmatrix{2 & 0 & -1\\0 & 2 & -1 \\ -1 & -1 & 3}$$
    The instructions that do that are "\pmatrix{2 & 0 & -1\\0 & 2 & -1 \\ -1 & -1 & 3}". Note the use of '&' as a separator, not a comma, and there is only one pair of curly brackets "{ }".

    Also, your eigenvalues read as ##\langle \frac{1}{\sqrt{6}} 1 , 1, -2 \rangle##, but you might have meant ##\frac{1}{\sqrt{6}} \langle 1,1,-2 \rangle##, which is very different. Using ##\vec{\mbox{ }}## does not work well for an array of more than about two characters in length, so ##\vec{v_1}## looks OK but ##\vec{v_1, v_2, v_3,v_4}## does not.
     
  14. Dec 30, 2017 #13
    Will have eigenvalues ##e^{3a+5}## with the same eigenvectors

    Thank you for your help
     
  15. Dec 30, 2017 #14

    PeroK

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    Is that ##exp(3a+5)## or ##exp(3a) + 5##?
     
  16. Dec 30, 2017 #15
    Oops sorry its meant to be ##exp(3a) + 5##
     
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