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

**Answer**

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∈Rn, 0<uTAu and 0<uTBu.

This means that 0<uTAu+uTBu or equivalently that 0<uT(A+B)u. ie. (A+B) has to be positive definite.

**Attribution***Source : Link , Question Author : Wis , Answer Author : usεr11852*