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A set of symmetric $n \times n$ matrices have $ \frac{1}{2}n^{2} - \frac{1}{2}n$ independent representations. But how do you get to this result? I understand that a general $n \times n$ matrix would have $n^{2}$ representations. How exactly does the constraint of symmetry lead to the piece above? Furthermore, how would I determine the number of representations of a symmetric traceless matrix?

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This is not representation theory........ –  user38268 Oct 20 '12 at 13:31
    
it's applications lie in representation theory. for example, the Lie algebra of O(n) can be represented by real nxn antisymmetric matrices, making dim(L(O(n))) = 1/2*n*(n-1). do you have any ideas at how to determine this dimension? that would be helpful. –  johndmalcolm Oct 20 '12 at 13:47
    
The lie algebra of O(n) has the addition constraint that the trace of an element sums to zero. –  user38268 Oct 20 '12 at 13:52

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