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Math Help - [SOLVED] Linear Algebra, trace similar matrices

  1. #1
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    [SOLVED] Linear Algebra, trace similar matrices

    a) Show that if A and B are n-by- n matrices with coefficients in an arbitrary field F, then Tr(AB) = Tr(BA).

    b) use (a) to show that if A and B are similar matrices, then Tr(A) = Tr(B).

    c) Show that the matrices of trace zero form a subspace of M{_n}(F) of dimension n{^2} - 1. [Hint: The mapping Tr: M{_n}(F) \rightarrow F is a linear transformation.]

    d) Prove that the subspace of M{_n}(F) defined in (c) is generated by the matrices AB - BA,\ A,\ B \in M{_n}(F).
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  2. #2
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    Quote Originally Posted by JJMC89 View Post
    a) Show that if A and B are n-by- n matrices with coefficients in an arbitrary field F, then Tr(AB) = Tr(BA).

    b) use (a) to show that if A and B are similar matrices, then Tr(A) = Tr(B).

    c) Show that the matrices of trace zero form a subspace of M{_n}(F) of dimension n{^2} - 1. [Hint: The mapping Tr: M{_n}(F) \rightarrow F is a linear transformation.]

    d) Prove that the subspace of M{_n}(F) defined in (c) is generated by the matrices AB - BA,\ A,\ B \in M{_n}(F).
    a) If A = (a_{ij}) and B = (b_{ij}) then \text{Tr}(AB) = \sum_{i,j=1}^na_{ij}b_{ji}, and \text{Tr}(BA) is given by the same formula.

    b) By a), \text{Tr}(P^{-1}(AP)) = \text{Tr}((AP)P^{-1}).

    c) Use the hint. The subspace consisting of matrices in M_n(F) with trace 0 is the kernel of the map in the hint.

    d) The identity A = \bigl(A-\tfrac1n\text{Tr}I_n\bigr) + \tfrac1n\text{Tr}I_n shows that every nn matrix can be written as the sum of a matrix with trace 0 (namely the expression in parentheses) and a multiple of the identity.

    Write E_{ij} for the matrix having a 1 in the (i,j)-position and zeros everywhere else. If i\ne j then E_{ij} = E_{ii}E_{ij} - E_{ij}E_{ii}. By a linear combination of such matrices you can construct any matrix with zeros along the diagonal as a linear combination of commutators. Next, E_{11}-E_{ii} = E_{1i}E_{i1} - E_{i1}E_{1i} for all i>1. Check that by forming linear combinations of such matrices you can form any diagonal matrix that is not a scalar multiple of the identity.

    That shows that any matrix with trace 0 can be expressed as sum of commutators.
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