It is well-know that for a Hermitian matrix the diagonal elements are majorized by the eigenvalues. Let M be a Hermitian matrix with eigenvalues \lambda_1, ..., \lambda_n and x a real eigenvector of M associated with the eigenvalue \lambda_k. Let y be any row vector in real field. It is not difficult to prove that M+xy has eigenvalues \lambda_1,... \lambda_{k-1}, \lambda_k+yx, \lambda_{k+1},... ,\lambda_n.
Simlar to Hermitian matrix, whether the majorization betweeen diagonal lements and eigenvalues of M+xy still holds.
Feeling is: to present a matrix as an identity matrix multiplied by the vector M+xy Since the vector M+xy can be presented in the form of a matrix so it's product and the identity matrix will be the matrix which is equal to the original vector M+xy This one can be presented as lambda_1,... \lambda_{k-1}, \lambda_k+yx, \lambda_{k+1},... ,\lambda_n. If you take lambda outside of the brackets you will have (k+yx...n+yx) If majorization holds for a vector it will also hold for a vector times identity matrix as matrix is only a number of elements arranged in columns and rows But I can be wrong on this Better to say something than nothing So Im saying
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