Eigenvalues of sum of a Hermitian matrix and a diagonal matrix

In summary, when forming a new matrix C by adding two matrices A and B with real eigenvalues, we cannot determine the exact values of the eigenvalues of C from the eigenvalues of A and B. However, we can determine that the sum of the eigenvalues of C is equal to the sum of the eigenvalues of A and B, and the smallest eigenvalue of C will be equal to or less than the average of the sum of the eigenvalues of A and B.
  • #1
peterlam
16
0
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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  • #2
c1+c2+...+cn=a1+a2+...+an+b1+b2+...+bn
min{c1,c2,...,cn} ≤ (a1+a2+...+an+b1+b2+...+bn)/n
 

What is a Hermitian matrix?

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. This means that the elements on the main diagonal are real numbers, and the elements below and above the diagonal are complex conjugates of each other.

What is a diagonal matrix?

A diagonal matrix is a square matrix where all non-diagonal elements are zero. This means that the matrix is represented by a diagonal line of numbers.

What is an eigenvalue?

An eigenvalue is a scalar value that represents the amount by which a vector is scaled when multiplied by a matrix. In other words, it is a number that tells us how the matrix is stretching or compressing a vector.

What is the significance of the sum of a Hermitian matrix and a diagonal matrix?

The sum of a Hermitian matrix and a diagonal matrix is significant because it is also a Hermitian matrix. This property allows us to simplify calculations and make predictions about the behavior of the matrix.

How do we find the eigenvalues of the sum of a Hermitian matrix and a diagonal matrix?

To find the eigenvalues of the sum of a Hermitian matrix and a diagonal matrix, we can use the fact that the sum of two Hermitian matrices is also a Hermitian matrix. This means that we can use the same methods for finding the eigenvalues of a Hermitian matrix, such as diagonalization or the characteristic polynomial method.

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