# Is the average of positive-definite matrices also positive-definite?

Is the average of multiple positive-definite matrices necessarily positive-definite or positive semi-definite? The average is element-wise average.

• Average is just sum followed by scaling. Is this true for each of those? – user541686 May 30 '17 at 1:29

Assume $A$ and $B$ are positive definite matrices for size $n$. By definition this means that for all $u \in R^n$, $0 < u^TAu$ and $0 < u^TBu$. This means that $0 < u^TAu + u^TBu$ or equivalently that $0 < u^T(A+B)u$. ie. $(A+B)$ has to be positive definite.