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Math Help - Matrices & char. polynomials.

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    MHF Contributor Also sprach Zarathustra's Avatar
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    Matrices & char. polynomials.

    Where I could find a nice proof of following theorem:

    A is matrix n x n over field F , similar to upper triangular matrix if and only if her characteristic polynomial can be factored into an expression with the form:
    (x-t_1)(x-t_2)...(x-t_n)
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Where I could find a nice proof of following theorem:

    A is matrix n x n over field F , similar to upper triangular matrix if and only if her characteristic polynomial can be factored into an expression with the form:
    (x-t_1)(x-t_2)...(x-t_n)
    det(A)=(a_{11}-\lambda)A_{11}+\sum_{i=2}^{n}a_{i1}A_{i1}

    (a_{11}-\lambda)A_{11}=(a_{11}-\lambda)(a_{22}-\lambda)...(a_{nn}-\lambda)

    =(-1)^n\lambda^n+...+(-1)^{n-1}\lambda^{n-1}

    p(0)=det(A)=\lambda_1\lambda_2...\lambda_n
    (-1)^{n-1}=tr(A)=\sum_{i=1}^{n}\lambda_i

    p(\lambda)=0 has exactly n solutions \lambda_1,...,\lambda_n

    p(\lambda)=(\lambda_1-\lambda)(\lambda_2-\lambda)...(\lambda_n-\lambda)

    p(0)=(\lambda_1)(\lambda_2)...(\lambda_n)=det(A)
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