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 $\displaystyle A,B\in\mathbb{R}^{n\times n}$ are positive definite then, $\displaystyle x^tAx>0,\;x^tBx>0$ for all $\displaystyle 0\neq x\in\mathbb{R}^n$ . This implies $\displaystyle x^t(A+B)x=x^tAx+x^tBx>0$ for all $\displaystyle 0\neq x\in\mathbb{R}^n$ .