Is the average of multiple positive-definite matrices necessarily positive-definite or positive semi-definite? The average is element-wise average.
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3$\begingroup$ Average is just sum followed by scaling. Is this true for each of those? $\endgroup$– user541686May 30, 2017 at 1:29
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2 Answers
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Yes, it is. jth asnwer is correct (+1) but I think you can get a much simple explanation with just basic Linear Algebra.
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.
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Of course. The set of positive definite matrices forms a cone, meaning it is closed under positive linear combinations and scaling.
