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Math Help - trouble with 4x4 matrices

  1. #1
    Junior Member
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    trouble with 4x4 matrices

    find all the eigenvalues and eigenvectors for matrix B

    b=
    2 0 1 1
    0 2 1 1
    1 1 2 0
    1 1 0 2

    i understand how to do 3x3 matrices but cant get my head around 4x4
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  2. #2
    Junior Member
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    i could solve the polynomial but that would take forever. is there a easier way.

    the polynomical i came up with is

    A^4 - 8A^3 + 24A^2 - 32A + 16 = 0
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  3. #3
    MHF Contributor

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    With all of those "2"s along the diagonal, \lambda always appear in " 2-\lambda". When I expand it, rather tham multiplying out powers of 2- \lambda, I left them as powers and got (2-\lambda)^4- (2-\lambda)^2= 0. Letting x= 2- \lambda, that equation becomes x^4- x^2= x^2(x^2- 1)= 0 which has solutions x= 0 (a double root), x= 1, and x= -1. Since x= 2-\lambda, \lambda= 2 is a double eigenvalue, and \lambda= 1 and \lambda= 3 are also eigenvalues.
    Last edited by HallsofIvy; February 7th 2010 at 07:47 AM.
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