Eigenvalues of hermitian matrix

In summary, the eigenvalues of a Hermitian matrix AH are related to the eigenvalues of A through their corresponding eigenvectors and a conditional equation involving their complex conjugates.
  • #1
fanxiu
3
0
1. Let AH be the hermitian matrix of matrix A, and how the eigenvalues of AH be related to eigenvalues of A?



3. what I have done is

equation no.1: (AH-r1*I) * x1 = 0,
And equation no.2: (A-r2*I) * x2 = 0

time no.1 both sides by x2H
((A*x2)H-r1*x2H)* x1 = 0

Then we have (conjugate (r2) -r1)*x2H*x1 = 0

only if x1 and x2H are not orthogonal vectors,
conjugate (r2) = r1

so i just reach this conditional conclusion. but i am not sure what else can be know about the eigenvalues of hermitian matrix
 
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  • #2
AH. The eigenvalues of a Hermitian matrix AH are related to the eigenvalues of A in the following way: if x1 and x2 are eigenvectors of A corresponding to eigenvalues r1 and r2 respectively, then x2H*x1 = 0 if and only if conjugate(r2) = r1. This means that the eigenvalues of AH are either the same as the eigenvalues of A or their complex conjugates, depending on whether the corresponding eigenvectors are orthogonal or not.
 

Related to Eigenvalues of hermitian matrix

1. What are eigenvalues of a Hermitian matrix?

Eigenvalues of a Hermitian matrix are the values that satisfy the characteristic equation of the matrix. They represent the possible values that a matrix can have when multiplied by a vector, and are essential in understanding the behavior of the matrix.

2. How are eigenvalues of a Hermitian matrix calculated?

The eigenvalues of a Hermitian matrix can be calculated by finding the roots of the characteristic equation, which is obtained by subtracting the identity matrix multiplied by a scalar from the original matrix and then taking the determinant.

3. What is the significance of eigenvalues in a Hermitian matrix?

Eigenvalues of a Hermitian matrix have several important properties, such as being real numbers, being related to the matrix's trace and determinant, and determining the matrix's diagonalizability. They are also used in solving systems of linear equations and in many other applications in mathematics, physics, and engineering.

4. How do the eigenvalues of a Hermitian matrix relate to its eigenvectors?

The eigenvalues of a Hermitian matrix are the coefficients of the corresponding eigenvectors. This means that when a Hermitian matrix is multiplied by its eigenvector, the resulting vector is a scaled version of the original eigenvector, with the scaling factor being the eigenvalue.

5. Can a Hermitian matrix have complex eigenvalues?

No, a Hermitian matrix can only have real eigenvalues. This is because the Hermitian property requires the matrix to be equal to its own conjugate transpose, which means that all of its eigenvalues must also be equal to their own complex conjugates.

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