# Thread: Is sum of positive definite matrices positive definite

1. ## 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.

2. ## Re: Is sum of positive definite matrices positive definite

Originally Posted by itpro
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 $0\neq x\in\mathbb{R}^n$ .

3. ## 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?

4. ## Re: Is sum of positive definite matrices positive definite

Originally Posted by itpro
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.

5. ## Re: Is sum of positive definite matrices positive definite

Originally Posted by itpro
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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