Diagonalize a non-hermitian matrix

In summary, the eigenvalues of a two dimensional matrix are the values of the function at the corresponding points.
  • #1
njuclean
3
0
I'm learning "the transformation optics" and the first document about this method is "Photonic band structures" ( Pendry, J. B. 1993). In this document, the transfer matrix T is non-hermitian, Ri and Li are the right and left eigenvectors respectively.

Pendry defined a unitary matrix S=[tex]\sum[/tex]RiLi, here RiLi is the outer product between the right eigenvector and the left eigenvector. Then T can be diagonalized through STS-1.

I remember that the sum of the outer product between the right eigenvector and the left eigenvector is the unit matrix, so i cannot understand how can the unitary matrix S diagonalize T. Who can tell me what's wrong in my understanding?
 
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  • #2
In general, S won't be the unit matrix. Consider it's operation on one of the left eigenvectors L_i. It will get transformed in the corresponding right eigenvector R_i. If the left and right eigenvectors do not conincide, S cannot be the unit matrix.
 
  • #3
DrDu said:
In general, S won't be the unit matrix. Consider it's operation on one of the left eigenvectors L_i. It will get transformed in the corresponding right eigenvector R_i. If the left and right eigenvectors do not conincide, S cannot be the unit matrix.

Thanks for your reply.

Are the right vectors of a non-hermitian matrix in general not the orthogonal and complete bases? I'm lack of the knowledge of linear algebra, so my question maybe too simple.
 
  • #4
The left or right eigenvectors form an orthogonal basis not for all possible non-Hermitean operators but only for a sub-class, the so-called "normal" operators. All normal operators can be expressed in the form A+iB where A and B are Hermitean operators.
Maybe you should try to work out the eigenvalues and eigenvectors for some very simple two dimensional matrices e.g. one with i and -i on the diagonal and zero on the outer diagonal.
 
  • #5
I'm glad to see your reply again.

The eigenvalues of the two dimensional matrix (a11=i, a12=0, a21=0, a22=-i) are [tex]\lambda[/tex]1=i and [tex]\lambda[/tex]2=-i. For [tex]\lambda[/tex]1, R1=(1 0)T and L1=(1 0). For [tex]\lambda[/tex]2, R2=(0 1)T and L2=(0 1). Thus [tex]\sum[/tex]RjLj=R1L1+R2L2=I.

I think it's necessary for me to read some books about the linear algebra. Anyway, thanks again for your help.
 
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Related to Diagonalize a non-hermitian matrix

1. What does it mean to diagonalize a non-hermitian matrix?

Diagonalizing a matrix is the process of finding a new basis for the matrix in which the matrix is represented as a diagonal matrix, with all off-diagonal elements being zero. A non-hermitian matrix is one that does not satisfy the condition of being equal to its own conjugate transpose. Therefore, diagonalizing a non-hermitian matrix means finding a new basis in which the matrix is represented as a diagonal matrix, but this may involve complex numbers.

2. Why is diagonalizing a non-hermitian matrix important?

Diagonalizing a matrix can simplify calculations and make certain properties of the matrix more transparent. For non-hermitian matrices, diagonalization is especially important because it allows us to identify the eigenvalues and eigenvectors, which are critical in understanding the behavior of the matrix.

3. Can any non-hermitian matrix be diagonalized?

No, not all non-hermitian matrices can be diagonalized. A matrix can only be diagonalized if it is diagonalizable, which means it has a full set of linearly independent eigenvectors. If a non-hermitian matrix does not have a full set of eigenvectors, it cannot be diagonalized.

4. How is diagonalization of a non-hermitian matrix different from a hermitian matrix?

The diagonalization process for a hermitian matrix is simpler because all hermitian matrices are diagonalizable. This means that all hermitian matrices have a full set of linearly independent eigenvectors, making the diagonalization process straightforward. For non-hermitian matrices, the diagonalization process may involve complex numbers and is only possible if the matrix has a full set of eigenvectors.

5. What are some applications of diagonalizing non-hermitian matrices?

Diagonalizing non-hermitian matrices has many applications in various fields of science and engineering, such as quantum mechanics, signal processing, and control theory. In quantum mechanics, diagonalization of non-hermitian matrices is used to find the energy levels and wavefunctions of quantum systems. In signal processing, diagonalization is used to analyze and manipulate signals in frequency domain. In control theory, diagonalization is used to simplify the analysis and design of control systems.

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