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Math Help - Symmetric Matrix

  1. #1
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    Symmetric Matrix

    The theorem state that the eigenvalues of a symmetric are real.

    [ 3 4 | has real eigenvalues 1 & 5 (verified) but why not symmetric?
    | 1 3 |


    Rgds
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by Chris0724 View Post
    The theorem state that the eigenvalues of a symmetric are real.

    [ 3 4 | has real eigenvalues 1 & 5 (verified) but why not symmetric?
    | 1 3 |


    Rgds
    This is logic !

    \text{Symmetric matrix } \implies \text{ Real eigenvalues}

    But this is different from \text{Real eigenvalues } \implies \text{ Symmetric matrix}.

    This is the converse of the implication. And is not always true.
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    Quote Originally Posted by Moo View Post
    Hello,

    This is logic !

    \text{Symmetric matrix } \implies \text{ Real eigenvalues}

    But this is different from \text{Real eigenvalues } \implies \text{ Symmetric matrix}.

    This is the converse of the implication. And is not always true.

    So in short, i will always get a Real eigenvalues from a symmetric matrix but Real eigenvalues does not meant that the matrix must be symmetric?

    Thanks!
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  4. #4
    Moo
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    Quote Originally Posted by Chris0724 View Post
    So in short, i will always get a Real eigenvalues from a symmetric matrix but Real eigenvalues does not meant that the matrix must be symmetric?

    Thanks!
    Exactly !

    The symmetry of a matrix is a sufficient but not necessary condition. If it was, you would have an equivalence, not an implication

    Necessary and sufficient condition - Wikipedia, the free encyclopedia
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