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Math Help - Algebraic Structures Matrix

  1. #1
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    Algebraic Structures Matrix

    I really need help, i need to investigate the properties of the set of matrices
    a -b
    b a :a,b belongs to real numbers

    I need to know
    • transformation properties
    • diagonalisation
    • group properties
    • subgroups
    • Isomorphism
    Any help would be fantastic!

    thanks!
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  2. #2
    Lord of certain Rings
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    You will have to be more specific than that to get useful answers. For the second part, I dont think you can diagonalize it(over reals) since its eigenvalues are always complex.

    What transformational property do you want? What group, subgroup properties are you looking at?
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  3. #3
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    I think that is why i am having the problem! That is exactly how the question is worded no extra information is given.
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  4. #4
    Moo
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    Hello,

    Quote Originally Posted by mrks1d View Post
    I think that is why i am having the problem! That is exactly how the question is worded no extra information is given.
    But it's not a problem to diagonalise it over \mathbb{C} o.O
    The eigenvectors will remain over \mathbb{R}

    \chi_A(\lambda)=(a-\lambda)^2+b^2=(a-\lambda-ib)(a-\lambda+ib)

    \lambda_1=a+ib
    \lambda_2=a-ib

    Then find the eigenvectors. To solve for X(x,y), representing the eigenvector, just remember that if a complex number is null, it means that its real part is zero and its imaginary part is zero..
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