Let us consider two arbitrary Hermitian square matrices $\mathbf{A,B}$ with the same dimension. Given $\mathbf{v}$ the eigenvector associated to the maximum eigenvalue of $\mathbf{A}$:



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> Are there any conditions appart from $\mathbf{A} = \mathbf{B}$ or $\mathbf{B} = \mathbf{v}\mathbf{v}^H$ so that it can be ensured that $\mathbf{B}$ has the same eigenvector associated to the maximum eigenvalue?
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Note that, in Dan Shemesh, Common eigenvectors of two matrices, Linear Algebra and its Applications, Volume 62, November 1984, Pages 11-18, ISSN 0024-3795, <http://dx.doi.org/10.1016/0024-3795(84)90085-5>. an arbitrary eigenvector was considered. I cross check the derivation, and I do not see how to promote the maximum eigenvector constraint.


Thank you in advance



Shared maximum eigenvector
This is true iff the largest eigenvalue of $A+B$ is the sum of that of $A$ and that of $B$.


