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Math Help - Algebra, Problems For Fun (19)

  1. #1
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    Algebra, Problems For Fun (19)

    A standard notation: By M_n(R) we mean the set of all n \times n matrices with entries from some set R.


    Problem: Let A \in M_n(\mathbb{C}). Suppose that \text{tr}(A)=\text{tr}(A^2)=\cdots = \text{tr}(A^n)=0. Prove that A^n=\bold{0}.


    Remark: The above result remains true if we replace \mathbb{C} with any field F of characteristic 0. It is not necessarily true if \text{char} F \neq 0. A counter-example is: A=\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \in M_2(\mathbb{F}_2).
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    Using the fact that the characteristic polynomial of an nxn matrix is a monic polynomial with coefficients that can be expressed in terms of it's trace and the traces of it's powers upto the power n we get that

     \chi_A(X) = \pm X^n as all the traces are 0.

    Then the result follows from the Cayley Hamilton Theorem.

    I couldn't think of a way of proving it that didnt rely on these two results but I'm sure there is a nice way.

    thanks, pomp.
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  3. #3
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    Quote Originally Posted by pomp View Post
    Using the fact that the characteristic polynomial of an nxn matrix is a monic polynomial with coefficients that can be expressed in terms of it's trace and the traces of it's powers upto the power n we get that

     \chi_A(X) = \pm X^n as all the traces are 0.

    Then the result follows from the Cayley Hamilton Theorem.

    I couldn't think of a way of proving it that didnt rely on these two results but I'm sure there is a nice way.

    thanks, pomp.
    let \lambda_1, \cdots , \lambda_n be the eigenvalues of A. we know that every matrix over \mathbb{C} is similar to some triangular matrix. so A=PBP^{-1}, for some triangular matrix B.

    then 0=\text{tr}(A^k)=\text{tr}(PB^kP^{-1})=\text{tr}(B^k)=\lambda_1^k + \cdots + \lambda_n^k, for all 1 \leq k \leq n, which gives us \lambda_1=\cdots=\lambda_n=0. therefore A^n=\bold{0}, by Cayley-Hamilton.
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