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Math Help - Positive Definite Proof

  1. #1
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    Positive Definite Proof

    We say a symmetric matrix A is positive definite if Ax dot x>0 for all x not zero, negative definite if Ax dot x<0 for all x not zero and positive semi-definite if Ax dot x >/= 0 for all x.

    Show that if A and B are positive definite, then so is A+B

    Show that if A is positive definite if and only if all its eigenvalues are positive.
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  2. #2
    Super Member girdav's Avatar
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    Re: Positive Definite Proof

    For the first question, write (A+B)x\cdot x in function of Ax\cdot x and Bx\cdot x. For the second question, take x an eigenvector for the eigenvalue \lambda.
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    Re: Positive Definite Proof

    Alright, i figured out the first part, but I'm a little unsure how you see to proceed for b?
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    Super Member girdav's Avatar
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    Re: Positive Definite Proof

    Let \lambda an eigenvalue for A, and x\neq 0 an eigenvector. Then Ax=\lambda x and x^tAx=\lambda \lVert x\rVert^2\geq 0 so \lambda \geq 0. Now, to see the converse, use spectral theorem.
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