Eigenvalues and eigenvectors of a matrix product

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The discussion centers on the relationship between the eigenvalues and eigenvectors of two non-negative nxn matrices, A and B, and their product C = A*B. It explores whether knowledge of the eigenvalues and eigenvectors of A and B can provide insights into those of C, particularly regarding the largest eigenvalue and associated eigenvector. A key point is that if A and B share an eigenvector, that vector will also be an eigenvector of both AB and BA with the product of their eigenvalues. However, the conversation raises the question of what can be inferred about C when A and B do not share eigenvectors, especially if their eigenvectors are similar but not identical. The discussion highlights the complexity of analyzing eigenvectors in the context of matrix products without the assumption of commutativity.
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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.
 
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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.
 
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?
 

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