# Is sum of positive definite matrices positive definite

• Jan 2nd 2012, 08:20 AM
itpro
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.
• Jan 2nd 2012, 08:36 AM
FernandoRevilla
Re: Is sum of positive definite matrices positive definite
Quote:

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 $0\neq x\in\mathbb{R}^n$ .
• Jan 2nd 2012, 08:40 AM
itpro
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?
• Jan 2nd 2012, 08:42 AM
Also sprach Zarathustra
Re: Is sum of positive definite matrices positive definite
Quote:

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.
• Jan 2nd 2012, 09:00 AM
FernandoRevilla
Re: Is sum of positive definite matrices positive definite
Quote:

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.