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Math Help - positive semi-definite matrix and determinant

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    positive semi-definite matrix and determinant

    Having a positive determinant is one of the main properties of a Postive Semi-Definite matrix. How do I prove it?
    Last edited by altave86; September 27th 2009 at 11:18 AM.
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    Quote Originally Posted by altave86 View Post
    Having a positive determinant is one of the main properties of a Postive Semi-Definite matrix. How do I prove it?
    let \lambda be an eigenvalue of A. then A\bold{x}=\lambda \bold{x}, for some \bold{x} \neq \bold{0}. thus 0 \leq \bold{x}^* A \bold{x}=\lambda \bold{x}^* \bold{x}=\lambda ||\bold{x}||^2 and so \lambda \geq 0. the result now follows because \det A is the product of the eigenvalues of A.
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