Constructing matrix with average eigenvalues

Question

Let $A,B$ be symmetric matrices with eigenvalues $a_1\le\dots\le a_n$ and $b_1\le\dots\le b_n$. Can we construct a matrix $C$ with eigenvalues $$\frac{a_1+b_1}{2},\dots,\frac{a_n+b_n}{2}?$$

The answer to this question is obviously yes, we can just take some diagonal matrix. But I am interested in whether there is some construction that does not require computing the eigenvalues of $A,B$?

Answer 1

You may take $C=\frac12(A+Q^TBQ)$, where $Q\in O(n,\mathbb R)$ is any global maximiser of $\operatorname{tr}(AQ^TBQ)$ (but it is another story to find $Q$).