General worked out solution for diagonalizing a 4x4 Hermitian matrix

[Amentia]

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!

 

[Paul Colby]

Doesn’t Singular Value Decomposition SVD do just this?

 

[Amentia]

Thank you for your answer. I have never used this method. I will have a look at it.

 

[mpresic3]

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

 

[Amentia]

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.

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.

 

[pasmith]

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.

 

[mpresic3]

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.