Eigenvalues of sum of a Hermitian matrix and a diagonal matrix


by peterlam
Tags: diagonal matrix, eigenvalue, hermitian
peterlam
peterlam is offline
#1
Mar2-11, 03:34 PM
P: 14
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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Some Pig
Some Pig is offline
#2
Mar3-11, 10:12 PM
P: 38
c1+c2+...+cn=a1+a2+...+an+b1+b2+...+bn
min{c1,c2,...,cn} ≤ (a1+a2+...+an+b1+b2+...+bn)/n


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