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Math Help - quadratic form and its eigenvalues

  1. #1
    Member garymarkhov's Avatar
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    quadratic form and its eigenvalues

    The question is: Consider the quadratic form x^TAx where A is a k x k matrix and x is a k x 1 vector. Show that if A is positive definite, then all its eigenvalues are positive.

    Not sure where to start.
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    Let \lambda be an eigenvalue of A and let x be an non-zero eigenvector corresponding to \lambda. What is x^TAx?
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    Quote Originally Posted by garymarkhov View Post
    The question is: Consider the quadratic form x^TAx where A is a k x k matrix and x is a k x 1 vector. Show that if A is positive definite, then all its eigenvalues are positive.

    Not sure where to start.

    Let \lambda\,\,and\,\,v be an eigenvalue of A and one of its corresponding eigenvectors, then:

    0<v^tAv=v^t(\lambda v)=\lambda v^tv \Longrightarrow \lambda>0\,\,\,as\,\,\,x^tx>0\,\,\,\forall\,\,\,ve  ctor\,\,\,x \neq 0

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