I have two correlation matrices A and B. They are:
- Real symmetric (with ones on the diagonal)
- Positive semi-definite (eigenvalues are $\ge 0$)
I want to try to prove that the average of these two matrices $C={1\over2}A + {1\over2}B$ still has the same properties.
It was a few years since my linear algebra courses and would like some help along the way here. I am assuming it's possible as I have tested it numerically and for my 5000 randomly generated correlation matrices the property holds (at least within machine precision).
Grateful for any kind of help!