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Math Help - Matrices and SLE

  1. #1
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    Matrices and SLE

    Let A = [1/2 -1/2; -1/2 1/2]

    Computer A^2 and A^3. What will A^n turn out to be?

    I computed A^2, how can I computer A^3? Is it possible to just multiply the product of A^2 with another A? That doesn't seem right to me though. And I have no idea how to find A^n. Thanks for the help!
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  2. #2
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    I found the answer in my book

    A^3 is equal to A*A^2 and A^n is equal to [a11^(n-1) a12^(n-1); a21^(n-1) a22^(n-1)] (general form)
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  3. #3
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    Hello,

    As A^3 = A.A.A, you can choose either to calculate AČ.A or A.AČ (the group of matrix is associative)
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  4. #4
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    You can find A^n by using diagnolization. Express MAM^{-1} = D where D is a diagnol matrix (which is abtained from the eigenvectors of A) then A = MDM^{-1} and so A^n = MD^nM^{-1}. Since D is diagnol it means the exponents D^n are simply the exponents of each of its diagnol entries.
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