I have two square matrices, say A and B. The matrices are both positive definite but not symmetric. The matrices have the same elements except for the elements on the main diagonal. The elements on the main diagonal of B are larger than or equal to the corresponding elements on A in magnitude. I want to show that the maximum magnitude eigenvalue of B is larger in magnitude than the maximum magnitude eigenvalue of A.
The Greshgorian disc theorem says that the eigenvalues of B are within circles with the same radius for A and B but the centers of the circles for B are larger. However, this doesn’t guarantee that the maximum eigenvalue of B is larger than the maximum eigenvalue of A in magnitude.
Is there anyway to show this?



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