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Finding the members of the Lie algebra of SO (n)

  1. Mar 4, 2015 #1
    1. The problem statement, all variables and given/known data
    Show that the members of the Lie algebra of SO(n) are anti-symmetric nxn matrices. To start, assume that the nxn orthogonal matrix R which is an element of SO(n) depends on a single parameter t. Then differentiate the expression:

    R.RT= I

    with respect to the parameter t, keeping in mind that I is a constant matrix. Then you must consider that the element M of the Lie algebra is defined as:

    M = (dR/dt) t=0

    And that R(0) is the identity matrix.

    2. Relevant equations
    (A.B)T = B T.A T

    3. The attempt at a solution
    d/dt[R(t).RT(t)] = 0

    I was introduced to linear algebra and group theory very recently and am having trouble doing any of the proofs that I am assigned for homework. I feel that this problem is probably easy, but it is surely not coming to me easily at all. . .

    Please help!
  2. jcsd
  3. Mar 4, 2015 #2
    I figured this out for the case of a 2x2 rotational matrix, but how would I generalize this for nxn matrices?
  4. Mar 4, 2015 #3


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    Homework Helper

    Use the product rule. Evaluate at ##t=0##. Remember a matrix ##A## being antisymmetric means ##A=(-A^T)##. You want to show ##R'(0)## is antisymmetric.
  5. Mar 8, 2016 #4
    Can you please provide complete solution? I have same question
  6. Mar 8, 2016 #5


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    Science Advisor
    Homework Helper

    No, that's not something we do. Try it and post your work if you want guidance. This isn't even hard if you give it some thought.
  7. Mar 9, 2016 #6
    Thanks. I solved it :)
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