Degrees of freedom - matrix diagonalization

This is a common technique in physics to simplify calculations and make them more manageable. In summary, bringing a symmetric matrix to diagonal form "consumes" degrees of freedom and reduces the number of independent components to only 4.
  • #1
Neitrino
137
0
Hi,
A symmetric 4x4 matrix has 10 independent components.
Let's say that matrix describes graviton h_mu_nu.
In general I can bring any symmetric matrix to diagonal form, so if can and I bring my h_mu_nu matrix to diagonal form where it has only 4 independent components ... than what happens to the rest 6 components...does this diagonalizing procedure (which in some sense means rotation in space-time and fixing my frame of reference ) "consumes" degrees of freedom ?

Could you please advise...

Thank you
 
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  • #2
Yes, the diagonalizing procedure does consume degrees of freedom. When a symmetric matrix is brought to a diagonal form, the 6 components that are not on the diagonal become redundant and can be discarded. This is essentially a rotation of the matrix in space-time, which means that any information about the original 6 components is lost.
 

1. What is meant by "degrees of freedom" in matrix diagonalization?

Degrees of freedom in matrix diagonalization refers to the number of independent variables needed to fully describe the system. In other words, it is the number of parameters that can vary without affecting the overall structure of the matrix.

2. How do you calculate the degrees of freedom for a given matrix?

The degrees of freedom for a matrix can be calculated by subtracting the number of constraints from the total number of variables. Constraints can include things like symmetry or orthogonality conditions imposed on the matrix.

3. Why is it important to consider degrees of freedom in matrix diagonalization?

Considering degrees of freedom in matrix diagonalization is important because it allows us to determine the minimum number of independent variables needed to fully describe the system. This can help us simplify and better understand complex systems.

4. How does increasing the number of degrees of freedom affect the diagonalization of a matrix?

Increasing the number of degrees of freedom makes the diagonalization of a matrix more challenging, as there are more independent variables to consider. However, it also allows for a greater range of possible solutions, which can be beneficial in certain situations.

5. Can the degrees of freedom for a matrix be negative?

No, the degrees of freedom for a matrix cannot be negative. It represents the minimum number of independent variables needed to fully describe the system, and therefore must be a positive value.

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