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Math Help - commutitive matrices

  1. #1
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    commutitive matrices

    hello
    please hellp me to prove that the only commutive matrices with aother matrices are the diagonal ???
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by loai View Post
    hello
    please hellp me to prove that the only commutive matrices with aother matrices are the diagonal ???
    That is not true. You should change diagonal by scalar. Try now.

    Fernando Revilla
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by loai View Post
    hello
    please hellp me to prove that the only commutive matrices with aother matrices are the diagonal ???
    Can you be more precise about what you are trying to prove?

    It is true that the only matrices that commute with all diagonal matrices are themselves diagonal matrices.

    But it is not true that the the only sets of commuting matrices are diagonal. For example, the matrices \begin{bmatrix}1&x\\0&1\end{bmatrix} all commute with each other.
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  5. #5
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    I think that loai was asking if it is true that the only matrices that commute with all other matrices are the diagonal matrices. That statement if true.
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    I think that loai was asking if it is true that the only matrices that commute with all other matrices are the diagonal matrices. That statement if true.
    I understood the problem in the same way, but the result is false. Choose:

    D=\begin{bmatrix}{1}&{0}\\{0}&{0}\end{bmatrix},\qu  ad A=\begin{bmatrix}{0}&{1}\\{0}&{0}\end{bmatrix}

    D is diagonal and however AD\neq DA

    In Drexel28's web there is a proof that a matrix M\in\mathbb{K}^{n\times n} conmutes with all matrices A\in \mathbb{K}^{n\times n} iff M is an scalar matrix.

    Fernando Revilla
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