Eigenvalues of sum of a Hermitian matrix and a diagonal matrixby peterlam Tags: diagonal matrix, eigenvalue, hermitian 

#1
Mar211, 03:34 PM

P: 14

Consider two matrices:
1) A is a nbyn Hermitian matrix with real eigenvalues a_1, a_2, ..., a_n; 2) B is a nbyn 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! 



#2
Mar311, 10:12 PM

P: 38

c_{1}+c_{2}+...+c_{n}=a_{1}+a_{2}+...+a_{n}+b_{1}+b_{2}+...+b_{n}
min{c_{1},c_{2},...,c_{n}} ≤ (a_{1}+a_{2}+...+a_{n}+b_{1}+b_{2}+...+b_{n})/n 


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