Solving Hermitian Matrix Homework: Pauli & σy

In summary, the conversation is about the hermiticity of Pauli matrices, specifically why the σx matrix is hermitian while the σy matrix is not. The individual is having trouble understanding why the σx matrix is hermitian and is able to prove the hermiticity of the σy matrix. They also question whether they are doing something wrong when taking the transpose and conjugate of the σx matrix. However, their matrix for σx is correct and they are reminded that since all elements are real, complex conjugation has no impact.
  • #1
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Homework Statement


I don't understand why the Pauli matrix σx is hermitian. Nonetheless, I am able to prove why the σy matrix is hermitian.


Homework Equations





The Attempt at a Solution


Whenever I do the transpose and then the conjugate I get the negative of σx instead. Am I doing something wrong or is this correct?
 
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  • #2
You're doing something wrong. Perhaps you don't have the correct matrix for [itex]\sigma_x[/itex].
 
  • #3
You're doing something wrong.
[tex]\sigma_x = \left( \begin{array} \\0&1\\ \\1&0\\ \end{array} \right)[/tex]

So when you transpose it it is the same. Since all of the elements are real, complex conjugation has no impact, so [tex] \sigma x = \sigma x ^\dagger[/tex]
 

What is a Hermitian matrix?

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

What is the role of Pauli matrices in solving Hermitian matrix homework?

Pauli matrices are a set of three 2x2 matrices that play a crucial role in solving Hermitian matrix homework. They are used to represent the spin of a particle in quantum mechanics and have properties that make them useful in solving equations involving Hermitian matrices.

What is the σy matrix in Hermitian matrix homework?

The σy matrix is one of the Pauli matrices and is used to represent the y-component of a particle's spin in quantum mechanics. It is also used in solving Hermitian matrix equations as it has properties that make it useful in these calculations.

How are Pauli matrices related to the concept of eigenvalues and eigenvectors?

Pauli matrices are closely related to the concept of eigenvalues and eigenvectors. In fact, they are used to find the eigenvalues and eigenvectors of Hermitian matrices. The eigenvalues and eigenvectors are important in understanding the behavior of quantum systems and play a crucial role in solving equations involving Hermitian matrices.

Are there any applications of solving Hermitian matrix homework in real-world scenarios?

Yes, solving Hermitian matrix equations has various applications in real-world scenarios. For example, they are used in quantum mechanics to understand the behavior of particles and in signal processing to analyze signals. They are also used in various fields of engineering and mathematics, such as in the study of vibrations, quantum computing, and data analysis.

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