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Math Help - Is sum of positive definite matrices positive definite

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    Is sum of positive definite matrices positive definite

    Is sum of two positive definite matrices positive definite and is there a
    proof or a theorem that shows that?

    Thank you.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Is sum of positive definite matrices positive definite

    Quote Originally Posted by itpro View Post
    Is sum of two positive definite matrices positive definite and is there a proof or a theorem that shows that?
    Yes, if A,B\in\mathbb{R}^{n\times n} are positive definite then, x^tAx>0,\;x^tBx>0 for all 0\neq x\in\mathbb{R}^n . This implies x^t(A+B)x=x^tAx+x^tBx>0 for all 0\neq x\in\mathbb{R}^n .
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    Re: Is sum of positive definite matrices positive definite

    This is great, Thank you!

    Is there a similar proof for products of two positive definite matrices?
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Is sum of positive definite matrices positive definite

    Quote Originally Posted by itpro View Post
    Is sum of two positive definite matrices positive definite and is there a
    proof or a theorem that shows that?

    Thank you.
    A hint (I belive) :

    If the matrix is positive definite, then all its eigenvalues are strictly positive.
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Re: Is sum of positive definite matrices positive definite

    Quote Originally Posted by itpro View Post
    This is great, Thank you! Is there a similar proof for products of two positive definite matrices?
    No, there isn't. The product of two definite positive matrices is definite positive if and only if the matrices commute.
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