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4x4 Matrix Eigenvalues and Eigenvectors

  1. Apr 30, 2015 #1
    1. The problem statement, all variables and given/known data
    I have 4 equations.

    3x+6y-5z-t=-8
    6x-2y+3z+2t=13
    4x-3y-z-3t=-1
    5x+6y-3z+4t=-6

    I have already solved this matrix using gauss elimination and found that x=1, y=2, z=5, t=-2

    Now the next part of the question asks to solve the matrix using eigenvalues and eigenvectors.

    2. Relevant equations
    Eigenvalues and Eigenvectors

    3. The attempt at a solution

    I have worked through to find the eigenvalues and received this equation

    0=L^4 - 4(L^3) - 30(L^2) - 162L + 537, where L is lander

    from this I found the Eigen values
    2.2522
    8.7456
    -3.4989-3.8757i
    -3.4989+3.8757i

    Which I am fairly confident are correct.
    However I am unsure about how to continue from here and how the corresponding eigenvectors will solve the matrix.

    Thanks Very Much
     
  2. jcsd
  3. Apr 30, 2015 #2

    HallsofIvy

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    Given that matrix A has eigenvalues [itex]\lambda_1[/itex], [itex]\lambda_2[/itex], [itex]\lambda_3[/itex], and [itex]\lambda_ 4[/itex], with corresponding eigenvectors [itex]v_1[/itex], [itex]v_2[/itex], [itex]v_3[/itex], and [itex]v_4[/itex], form the matrix P having those eigenvectors as columns and diagonal matrix D having the eigenvalues on its diagonal. Then [itex]A= PDP^{-1}[/itex]. The equation Ax= y is the same as [itex]PDP^{-1}x= y[/itex] and then [itex]DP^{-1}x= P^{1}y[/itex], [itex]P^{-1}x= D^{-1}P^{-1}y[/itex], and, finally, [itex]x= PD^{-1}P^{-1}y[/itex]. It is relatively easy to find [itex]P^{-1}[/itex] and [itex]D^{-1}[/itex] is just the diagonal matrix with the reciprocals of the diagonal elements of D on its diagonal.
     
    Last edited: May 2, 2015
  4. Apr 30, 2015 #3

    SteamKing

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    Generally, the Greek letter lambda, λ, (not "lander") is used for the unknown eigenvalues, thus the characteristic equation is written:

    λ4 - 4λ3 - 30λ2 - 162λ + 537 = 0

    They are.
     
  5. Apr 30, 2015 #4
    thanks, my bad. for some reason everyone I work with calls it lander instead of lambda
     
  6. Apr 30, 2015 #5

    HallsofIvy

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    If you are anywhere near Boston, Massachusetts, they may be saying "lamb-der"!
     
  7. May 2, 2015 #6

    Ray Vickson

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    Does your question go on to explain what it means when it says "solve the matrix"? There are many things you can do with eigenvalues/eigenvectors; youcan use them to solve numerous, varied types of "problems", but I cannot figure out what problem the question is now asking you to solve.
     
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