Thank you very much! Thanks to Prof. Breuer for so much trouble.
Both of you are right for Hermitian cases, indeed it is Weyl's min-max theorem.
However, for non-Hermitian matrices, the problem seems still open.
My conjecture is: The eigenvalues of A+B locate inside the convex hull formed by all λ(A)+λ(B).
My context comes from another problem in control theory, see
https://www.researchgate.net/post/Can_anyone_help_me_with_the_stabilization_of_marginally_stable_matrix?_tpcectx=qa_overview_following&_trid=53fafda3cf57d772028b4687_
I am a little surprised about the precise question here: Are you talking about matrices with real eigen values only? Or do you assume positive definite?
For hermitian matrices a lot of work was done in recent years, have to check the literature for this.
Also I'd like to see reference for the theorem (Weyl?) mentioned by Peter T Breuer.
Many thanks, however, I am still a little surprised. Assume we have two diagonal matrices A and B, then its obvious how the eigen values of tA + (1-t)B change. Now we take two commuting matrices A and B, then they are simultaneously diagonisable (provided both A and B are). So the same argument works as well.
Finally take A to be the identity matrix, then we get Weyl's theorem, since A is commuting with any B?
If A and B are not commuting, we have seen a counter example above.
So still, what is the precise problem here?
if matrices A and B are positive semidefinite ,then εA+(1-ε)B convex
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