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Thread: 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

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

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