Eigenvalues and eigenvectors of a matrix product

[Leo321]

We have two nxn matrices with non-negative elements, A and B.
We know the eigenvalues and eigenvectors of A and B.
Can we use this information to say anything about the eigenvalues or eigenvectors of C=A*B?
The largest eigenvalue of C and the associated eigenvector are of particular interest.
So can anything be said about C? Even a weak inequality may be useful. Are there particular sets of A and B, for which we can say something?
We can't assume however that the matrices commute.

 

[HallsofIvy]

We can provided A and B share an eigenvector. If v is an eigenvector of A with eigenvalue $\lambda_A$ and also an eigenvector of B with eigenvalue $\lambda_B$ then
$$ABv= A(Bv)= A(\lambda_Bv)= \lambda_BAv= \lambda_B(\lambda_Av)= (\lambda_B\lambda_A)v$$
and
$$BAv= B(Av)= B(\lambda_Av)= \lambda_ABv= \lambda_A(\lambda_Bv)= (\lambda_A\lambda_B)v$$

That is, if v is an eigenvector of both A and B, with eigenvalues $\lambda_A$ and $\lambda_B$ respectively, then it is also an eigenvector of both AB and BA with eigenvalue $\lambda_A\lambda_B$.

 

[Leo321]

The question is what can we know about the product if there are no shared eigenvectors. What happens for example if the eigenvectors are close, but not the same?
Is there anything we can say about the eigenvectors of the product based on the eigenvectors and eigenvalues of A and B?