Eigenvalues of sum of a Hermitian matrix and a diagonal matrix

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The discussion focuses on the eigenvalues of a new matrix C formed by adding a Hermitian matrix A and a diagonal matrix B. It is established that while the sum of the eigenvalues of C equals the sum of the eigenvalues of A and B, the individual eigenvalues of C cannot be directly determined from those of A and B. However, it is noted that the minimum eigenvalue of C can be bounded by the average of the eigenvalues of A and B. Specifically, the minimum eigenvalue of C is less than or equal to the average of the sums of the eigenvalues of A and B. This analysis provides insight into the relationships between the eigenvalues of combined matrices.
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Consider two matrices:
1) A is a n-by-n Hermitian matrix with real eigenvalues a_1, a_2, ..., a_n;
2) B is a n-by-n diagonal matrix with real eigenvalues b_1, b_2, ..., b_n.

If we form a new matrix C = A + B, can we say anything about the eigenvalues of C (c_1, ..., c_n) from the eigenvalues of A and B? Can we determine c_1, ..., c_n from a_1, ..., a_n, b_1, ..., b_n? If not, can we just determine the smallest eigenvalue of C from A and B?

Thank you!
 
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c1+c2+...+cn=a1+a2+...+an+b1+b2+...+bn
min{c1,c2,...,cn} ≤ (a1+a2+...+an+b1+b2+...+bn)/n
 

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