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Math Help - Uniqueness of Trace

  1. #1
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    Uniqueness of Trace

    Here's the problem:

    Let f be a function defined on the set of n \times n matrices with entries from the field F such that

    f(A + B) = f(A) + f(B),

    f(\lambda A) = \lambda f(A),

    f(AB) = f(BA).

    Prove that there is an element \alpha_0 \in F such that f(A) = \alpha_0 \text{trace} (A).

    I know that it can be shown that f has the same value for all matrices in a similarity class, but I have no idea where to go from there. Any insight is appreciated!
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  2. #2
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    Quote Originally Posted by syme.gabriel View Post
    Let f be a function defined on the set of n \times n matrices with entries from the field F such that

    f(A + B) = f(A) + f(B),

    f(\lambda A) = \lambda f(A),

    f(AB) = f(BA).

    Prove that there is an element \alpha_0 \in F such that f(A) = \alpha_0 \text{trace} (A).
    Write E_{ij} for the matrix with a 1 (the identity element of F) in the (i,j)-position and zeros everywhere else. If i≠j then E_{ii}E_{ij} = E_{ij} and E_{ij}E_{ii} = 0, from which it follows that f(E_{ij})=0. Also, E_{ij}E_{ji} = E_{ii} and E_{ji}E_{ij} = E_{jj}, so that f(E_{ii}) = f(E_{jj}). So if f(E_{11}) = \alpha_0 then f(E_{ii}) = \alpha_0 for all i. Thus f(E_{ij}) = \alpha_0\text{tr}(E_{ij}) for each matrix unit E_{ij}. Since these elements linearly generate the whole of M_n(F), the result follows.
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