Sigma matrices question Group theory

In summary, the question is about why every 2x2 Hermitian matrix satisfies a certain equation, and it may not be obvious to the person asking. However, it is clear from the textbook's expression (5.27) that Hermitian matrices have real diagonal elements and complex conjugate off-diagonal elements. By selecting values for the variables, any Hermitian matrix can be obtained. There may not be a lengthy calculation required to show this.
  • #1
helpcometk
75
0

Homework Statement


I have read the following text in a textbook(look the attaxhement) ,and i have a
simple question .WHY every 2x2 hermitian matrix would have to satisfy this Equation.It is not obvious to me why.Does anyone know the answer?
The textbook stops there without giving any answer.Is it something obvious?
Is there any lengthy calculation to show this?


Homework Equations





The Attempt at a Solution

 

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  • #2
helpcometk said:

Homework Statement


I have read the following text in a textbook(look the attaxhement) ,and i have a
simple question .WHY every 2x2 hermitian matrix would have to satisfy this Equation.It is not obvious to me why.Does anyone know the answer?
The textbook stops there without giving any answer.Is it something obvious?
Is there any lengthy calculation to show this?


Homework Equations





The Attempt at a Solution


It's sort of obvious just looking at the expression (5.27). Hermitian matrices have real diagonal elements and the two off-diagonal elements are complex conjugates. By selecting values of x0,x1,x2,x3 you can get any such matrix.
 
  • #3
Yes its sort of obvious that I am stupid.
Thanks
 
  • #4
helpcometk said:
Yes its sort of obvious that I am stupid.
Thanks

Hey, I miss the obvious too sometimes. Doesn't make you stupid.
 
  • #5


The equation in question, known as the "Sigma matrices equation," is a fundamental property of 2x2 Hermitian matrices in group theory. It states that the product of any two 2x2 Hermitian matrices is equal to the sum of the two matrices multiplied by the identity matrix. This equation is not obvious and requires a proof, which can be found in advanced textbooks on group theory or linear algebra. It is not something that can be easily explained in a short response, as it involves complex mathematical concepts and calculations. However, it is a well-established property of Hermitian matrices and plays a crucial role in understanding their properties and applications in various fields of science and engineering. If you would like to understand the proof of this equation, I suggest consulting a textbook or seeking help from a mathematician or group theory expert.
 

FAQ: Sigma matrices question Group theory

What are Sigma matrices in group theory?

Sigma matrices are a set of matrices used in the representation theory of finite groups. They are square matrices with entries that are either 0 or 1, and are used to encode the group structure and symmetry of a given group.

How are Sigma matrices related to group representations?

Sigma matrices are used to construct group representations, which are mappings from a group to a set of matrices that preserve the group structure. In particular, the entries of the Sigma matrices determine the basis for the representation.

What is the significance of the 0 and 1 entries in Sigma matrices?

The 0 and 1 entries in Sigma matrices correspond to the absence or presence of certain group elements in the basis for the representation. This allows for the representation to accurately reflect the group's symmetry and structure.

How are Sigma matrices used in character theory?

Sigma matrices are used in character theory to calculate the characters of group representations. The character of a representation is a function that assigns a value to each group element and is used to identify and classify representations.

Can Sigma matrices be used for infinite groups?

Yes, Sigma matrices can be used for both finite and infinite groups. However, for infinite groups, the matrices may have infinite dimensions, making computations more challenging.

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