General worked out solution for diagonalizing a 4x4 Hermitian matrix

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Discussion Overview

The discussion revolves around the diagonalization of a general 4x4 Hermitian matrix, exploring methods, challenges, and specific cases that may simplify the process. Participants inquire about resources and techniques, including the potential use of Singular Value Decomposition (SVD) and computer algebra systems.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks a worked-out solution for diagonalizing a general 4x4 Hermitian matrix and questions the existence of resources for this calculation.
  • Another participant suggests that Singular Value Decomposition (SVD) may be applicable for diagonalization.
  • A participant notes that solving the characteristic equation for a general 4x4 Hermitian matrix is complex and may require extensive computation, potentially filling several notebooks.
  • There is mention of using computer algebra systems like MATLAB, Mathematica, or Maple to assist in the diagonalization process, especially for specific cases.
  • One participant reflects on their attempts to simplify the problem by hand, noting that the Hermitian property does not necessarily simplify finding eigenvalues or eigenvectors.
  • Another participant points out that if the matrix is block diagonal, it can simplify the diagonalization process significantly, reducing it to smaller matrices.

Areas of Agreement / Disagreement

Participants generally agree on the complexity of diagonalizing a general 4x4 Hermitian matrix and the potential utility of computer algebra systems. However, there is no consensus on specific simplifications or methods that universally apply, as some participants highlight the challenges while others suggest possible approaches.

Contextual Notes

Participants express uncertainty regarding the simplification of the diagonalization process, particularly in the general case versus specific cases like block diagonal matrices. The discussion reflects varying levels of familiarity with computational tools and methods.

Amentia
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Hello,

I am looking for a worked out solution to diagonalize a general 4x4 Hermitian matrix. Is there any book or course where the calculation is performed? If not, does this exist for the particular case of a traceless matrix? Thank you!
 
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Doesn’t Singular Value Decomposition SVD do just this?
 
Thank you for your answer. I have never used this method. I will have a look at it.
 
The general solution with "general" coefficients in the matrix would likely fill up several notebooks. Just solving for the solution to the characteristic equation, a fourth degree equation, done in MATLAB with the symbolic toolbox goes on for about 10 pages of dense combinations of coefficients.

Your best bet to see the solution would be to use a computer algebra routine, like the symbolic toolbox in MATLAB, mathematica, Maple, or something like that. Even if you could solve it in the general case, how would you use the general solution. Your best bet might be to write a program to solve for the diagonalization for user chosen specific cases for your supplied coefficients
 
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mpresic3 said:
The general solution with "general" coefficients in the matrix would likely fill up several notebooks. Just solving for the solution to the characteristic equation, a fourth degree equation, done in MATLAB with the symbolic toolbox goes on for about 10 pages of dense combinations of coefficients.

I tried by hand at first just to see if there was some kind of simplification due to the fact that the matrix is Hermitian, but it seemed still to be complicated. Thank you for confirming it.

mpresic3 said:
Your best bet to see the solution would be to use a computer algebra routine, like the symbolic toolbox in MATLAB, mathematica, Maple, or something like that. Even if you could solve it in the general case, how would you use the general solution. Your best bet might be to write a program to solve for the diagonalization for user chosen specific cases for your supplied coefficients

I tried with Mathematica as well but I am barely using it, so I was not confident that I had simplified correctly the calculation...

So there are not even some specific cases where it is known to be simple? In one limiting case, I had block diagonal matrices and it was easy.
 
Amentia said:
I tried by hand at first just to see if there was some kind of simplification due to the fact that the matrix is Hermitian, but it seemed still to be complicated. Thank you for confirming it.

The matrix being Hermitian just guarantees that the matrix is diagonalizable and the eigenvalues are real. It doesn't simplify the process of finding eigenvalues or eigenvectors.
 
If the 4x4 is block diagonal, this simplifies the diagonalization to either a 3x3, and a 1x1, (trivial), or two 2x2's, a 2x2 diagonalization can probably done on a sheet of paper. A 3x3 "general" diagonalization would still be very difficult.
 

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