Assume that A, B are positive n by n matrices and the rank of B is 1, B=xx*. If the eigenvalues of A are a_1≥a_2≥...≥a_n, and x is not the eigenvector of A, then there are d_i≥0 such that eigenvalue of A+B are a_1+d_1, a_2+d_2,...,a_n+d_n. Is it true? Are d_is non-negative?
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