Generic Form of Hermitian Matrix

In summary, a generic Hermitian matrix is a square matrix that is equal to its own conjugate transpose. This means that the entries satisfy the condition \( A_{ij} = \overline{A_{ji}} \), where \( \overline{A_{ji}} \) denotes the complex conjugate of the entry. Hermitian matrices have real eigenvalues, and their eigenvectors corresponding to distinct eigenvalues are orthogonal. They play a significant role in various fields, including quantum mechanics and linear algebra, due to their properties related to symmetry and spectral decomposition.
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thatboi
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I am reading the following thesis: https://www.kip.uni-heidelberg.de/Veroeffentlichungen/download/6387/pdf-6387.pdf
Specifically, I am confused about equation (2.5), where they give the generic form of the matrix ##\mathcal{M}## due to the Hermiticity of ##\mathcal{H}## and the commutation relation (2.4). I am confused about why the bottom right element is ##\bar{A}##. I'm sure this is related to the commutation relation but I'm confused as to how they enter into the picture. ##\mathcal{H}^{\dagger} = (\mathcal{M}a)^{\dagger}(a^{\dagger})^{\dagger} = a^{\dagger}\mathcal{M}^{\dagger}a## so where do the commutation relations come from or what step did I skip?
 
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  • #2
Yes, something is missing, because I can take ##N=1## and ##M=\begin{pmatrix} 0&1\\1&2 \end{pmatrix}##, which make ##H=a^\dagger a^\dagger + aa+2aa^\dagger##, which is hermitian.
 
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FAQ: Generic Form of Hermitian Matrix

What is a Hermitian matrix?

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. In other words, a matrix \(A\) is Hermitian if \(A = A^\dagger\), where \(A^\dagger\) is the conjugate transpose of \(A\).

What is the generic form of a Hermitian matrix?

The generic form of a Hermitian matrix is such that the elements on the main diagonal are real numbers, and the elements off the main diagonal are complex numbers where the element in the \(i\)-th row and \(j\)-th column is the complex conjugate of the element in the \(j\)-th row and \(i\)-th column. Mathematically, if \(A\) is a Hermitian matrix, then \(A_{ij} = \overline{A_{ji}}\).

What are the properties of Hermitian matrices?

Hermitian matrices have several important properties: (1) All eigenvalues of a Hermitian matrix are real. (2) Eigenvectors corresponding to different eigenvalues are orthogonal. (3) Hermitian matrices are unitarily diagonalizable, meaning they can be written as \(A = U \Lambda U^\dagger\), where \(U\) is a unitary matrix and \(\Lambda\) is a diagonal matrix.

How are Hermitian matrices used in quantum mechanics?

In quantum mechanics, Hermitian matrices (also known as Hermitian operators) represent observable physical quantities, such as energy, position, and momentum. The real eigenvalues of these matrices correspond to the possible measurement outcomes, and the eigenvectors represent the states of the system.

How do you verify if a given matrix is Hermitian?

To verify if a given matrix \(A\) is Hermitian, you need to check if it is equal to its conjugate transpose. This involves taking the transpose of the matrix and then taking the complex conjugate of each element. If the resulting matrix is identical to the original matrix, then \(A\) is Hermitian.

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