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.

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 uRn, 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

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