Diagonalizing Hermitian matrices with adjoint representation

In summary, the conversation discusses the adjoint action of a matrix that diagonalizes a Hermitian ##N \times N## matrix and its relationship to the orthogonal group. It also raises questions about the elements of the matrix and their orthogonality.
  • #1
Luck0
22
1
Suppose I have a hermitian ##N \times N## matrix ##M##. Let ##U \in SU(N)## be the matrix that diagonalizes ##M##: ##M = U\Lambda U^\dagger##, where ##\Lambda## is the matrix of eigenvalues of ##M##. This transformation can be considered as the adjoint action ##Ad## of ##SU(N)## over its algebra ##\mathfrak{su}(N)##, so if I consider the generators ##\{t_a\}## of ##\mathfrak{su}(N)##, I can write expansions: ##Ad(U)t_a = Ut_aU^\dagger = \sum_b\Omega_{ab}(U)t_b##, ##M = \sum_a m_at_a##, ##\Lambda = \sum_a \lambda_a t_a##, where ##\Omega_{ab} \in O(N^2-1)##, the orthogonal group, and the diagonalization can be written as ##M = \sum_{a,b}\lambda_b\Omega_{ab}t_a##.

Note that since ##\Lambda## is diagonal, the generators in the expansion ##\Lambda = \lambda_a t_a## must also be diagonal, i.e., they must span the Cartan subalgebra of ##\mathfrak{su}(N)##, meaning that some of the coefficients ##\lambda_a## must be zero. This means that some of the columns of each matrix ##\Omega## will not enter in the sum ##\sum_{a,b}\lambda_b\Omega_{ab}t_a##. My question is, if I look at ##M = \sum_{a,b}\lambda_b\Omega_{ab}t_a## as a change of coordinates, can I still consider ##\Omega## as an element of ##O(N^2-1)##? Clearly, if I write the elements of ##\Omega## that appear in the sum in matrix form, it will be a rectangular matrix, but I don't know if I can complete it with zeros until I get a square matrix, and if I can, I can't see what happens to orthogonality.

Thanks in advance
 
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  • #2
It is difficult to follow you. I think it will be clearer in a coordinate free wording. Complementations with zeros (or ones) are usually a hidden embedding. So what is here to be embedded where? And why are some ##\lambda_a=0\,?##
 

Related to Diagonalizing Hermitian matrices with adjoint representation

1. What is the adjoint representation of a Hermitian matrix?

The adjoint representation of a Hermitian matrix is a linear transformation that maps the matrix onto its complex conjugate transpose. It is also known as the conjugate transpose or Hermitian conjugate.

2. Why is it important to diagonalize Hermitian matrices with adjoint representation?

Diagonalizing Hermitian matrices with adjoint representation allows for easier calculation of eigenvalues and eigenvectors, which are important in many applications such as quantum mechanics and signal processing. It also simplifies the matrix and makes it more interpretable.

3. How do you diagonalize a Hermitian matrix with adjoint representation?

To diagonalize a Hermitian matrix with adjoint representation, you can use the spectral theorem which states that a Hermitian matrix can be diagonalized by a unitary matrix. This involves finding the eigendecomposition of the matrix and using the resulting eigenvectors as columns in the unitary matrix.

4. What are the properties of a Hermitian matrix with adjoint representation?

A Hermitian matrix with adjoint representation has the property that its eigenvalues are all real and its eigenvectors are all orthogonal. It is also self-adjoint, meaning that it is equal to its own adjoint.

5. Can any matrix be diagonalized with adjoint representation?

No, not all matrices can be diagonalized with adjoint representation. Only square matrices that are Hermitian can be diagonalized in this way. Other types of matrices, such as non-Hermitian or non-square matrices, may require different methods of diagonalization.

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