# What can I infer about the eigenvalues of the sum of two matrices with known eigendecompositons

I have two matrices and I know the eigenvalues and eigenvectors of both, it also happens to be the case that they share the same set of eigenvalues. I now need to sum the two matrices and take the average, and finally, I need to compute the eigenvalues of the average matrix. Is there any shortcut I can take to perform the last step without digaonlization?

Consider this example: let $$A=\begin{pmatrix}1 & 0 \\ 0 & 2\end{pmatrix}$$ with eigenvalues $$1$$ and $$2$$, and let $$B$$ be the same matrix, but in another basis, say rotated by $$\varphi$$: $$B=\begin{pmatrix}1+\sin(\varphi)^2 & \sin(\varphi)\cos(\varphi) \\ \sin(\varphi)\cos(\varphi) & 1+\cos(\varphi)^2\end{pmatrix}$$. You can check that $$B$$ has eigenvalues $$1$$ and $$2$$ corresponding to eigenvectors $$\begin{pmatrix}\cos(\varphi) \\ -\sin(\varphi)\end{pmatrix}$$ and $$\begin{pmatrix}\sin(\varphi) \\ \cos(\varphi)\end{pmatrix}$$.
Now, the characteristic polynomial of $$\frac{A+B}{2}$$ is $$\lambda^2-3\lambda+2+\frac{\sin(\varphi)^2}{4}$$, with discriminant $$D=1-\sin(\varphi)^2$$, so eigenvalues are $$\frac{3\pm\sqrt{1-\sin(\varphi)^2}}{2}$$, which can be anything between $$1$$ and $$2$$.
Note that for $$\varphi=0$$ we get $$A=B$$, while for $$\varphi=\frac{\pi}{2}$$ we get $$B=\begin{pmatrix}2 & 0 \\ 0 & 1 \end{pmatrix}$$ and $$\frac{A+B}{2}=\frac{3}{2}I$$.