On the diagonalization of matrices and strange results

Question

I'm trying to implement the following calculation in mathematica

$$U^{(0)}(\mu,m) = V\left(\left[\frac{\alpha_S(m)}{\alpha_S(\mu)}\right]^{\frac{\gamma^{(0)}}{2\beta_0}}\right)_DV^{-1}$$

where $\gamma^{(0)}$ is a given $10\times 10$ matrix which depends on three parameters, $\beta_0$ is a function which depends on two parameters and $V$ is the matrix which diagonalizes $\gamma^{(0)T}$

$$\gamma_D^{(0)} = V^{-1}\gamma^{(0)}V$$

In mathematica I've implemented the following code

GammaS0[Nc_, Nu_, Nd_] := ({
   {-6/Nc, 6, 0, 0, 0, 0, 0, 0, 0, 0},
   {6, -6/Nc, -2/(3 Nc), 2/3, -2/(3 Nc), 2/3, 0, 0, 0, 0},
   {0, 0, -22/(3 Nc), 22/3, -4/(3 Nc), 4/3, 0, 0, 0, 0},
   {0, 0, 
    6 - (2 (Nd + Nu))/(3 Nc), -6/Nc + 
     2 (Nd + Nu)/3, (-2 (Nd + Nu))/(3 Nc), 2 (Nd + Nu)/3, 0, 0, 0, 
    0},
   {0, 0, 0, 0, 6/Nc, -6, 0, 0, 0, 0},
   {0, 0, -2 (Nd + Nu)/(3 Nc), 
    2 (Nd + Nu)/3, -(2 (Nd + Nu))/(3 Nc), -6 (-1 + Nc^2)/(Nd + Nu) + 
     2 (Nd + Nu)/3, 0, 0, 0, 0},
   {0, 0, 0, 0, 0, 0, 6/Nc, -6, 0, 0},
   {0, 0, -2 (Nu - Nd/2)/(3 Nc), 
    2 (Nu - Nd/2)/3, -2 (Nu - Nd/2)/(3 Nc), 2 (Nu - Nd/2)/3, 
    0, -6 (-1 + Nc^2)/Nc, 0, 0},
   {0, 0, 2/(3 Nc), -2/3, 2/(3 Nc), -2/3, 0, 0, -6/Nc, 6},
   {0, 0, -2 (Nu - Nd/2)/(3 Nc), 
    2 (Nu - Nd/2)/3, -2 (Nu - Nd/2)/(3 Nc), 2 (Nu - Nd/2)/3, 0, 0, 
    6, -6/Nc}
  })
GammaE0[Nc_, Nu_, Nd_] := ( {
   {-8/3, 0, 0, 0, 0, 0, 16 Nc/27, 0, 16 Nc/27, 0},
   {0, -8/3, 0, 0, 0, 0, 16/27, 0, 16/27, 0},
   {0, 0, 0, 0, 0, 0, -16/27 + 16 Nc (Nu - Nd/2)/27, 
    0, -88/27 + 16 Nc (Nu - Nd/2)/27, 0},
   {0, 0, 0, 0, 0, 0, -16 Nc/27 + 16 (Nu - Nd/2)/27, 
    0, -16 Nc/27 + 16 (Nu - Nd/2)/27, -8/3},
   {0, 0, 0, 0, 0, 0, 8/3 + 16 Nc (Nu - Nd/2)/27, 0, 
    16 Nc (Nu - Nd/2)/27, 0},
   {0, 0, 0, 0, 0, 0, 16 (Nu - Nd/2)/27, 8/3, 16 (Nu - Nd/2)/27, 0},
   {0, 0, 0, 0, 4/3, 0, 4/3 + 16 Nc (Nu + Nd/4)/27, 0, 
    16 Nc (Nu + Nd/4)/27, 0},
   {0, 0, 0, 0, 0, 4/3, 16 (Nu + Nd/4)/27, 4/3, 16 (Nu + Nd/4)/27, 
    0},
   {0, 0, -4/3, 0, 0, 0, 8/27 + 16 Nc (Nu + Nd/4)/27, 
    0, -28/27 + 16 Nc (Nu + Nd/4)/27, 0},
   {0, 0, 0, -4/3, 0, 0, 8 Nc/27 + 16 (Nu + Nd/4)/27, 0, 
    8 Nc/27 + 16 (Nu + Nd/4)/27, -4/3}
  } )
Gamma0[Nc_, Nu_, Nd_] := GammaS0[Nc, Nu, Nd] + GammaE0[Nc, Nu, Nd]

which are the definition of the $\gamma$ matrices, and then

V[Nc_, Nu_, Nd_] := 
 Normalize /@ 
  Eigenvectors[Transpose[Gamma0[Nc, Nu, Nd]], Cubics -> True]
Gamma0D[Nc_, Nu_, Nd_] := 
 DiagonalMatrix[Eigenvalues[Gamma0[Nc, Nu, Nd]]]
U0[mu_, m_] := 
 If[mu >= mt, 
  MatrixExp[
   Log[AlphaS[mu, Lambda, 6, Nc]/AlphaS[m, Lambda, 6, Nc]] Gamma0[Nc, 
      3, 3]/(2 Beta0[6, Nc])],
            If[mb <= mu < mt, 
   MatrixExp[
    Log[AlphaS[mu, Lambda, 5, Nc]/AlphaS[m, Lambda, 5, Nc]] Gamma0[Nc,
        2, 3]/(2 Beta0[5, Nc])],
            If[mc <= mu < mb, 
    MatrixExp[
     Log[
       AlphaS[mu, Lambda, 4, Nc]/AlphaS[m, Lambda, 4, Nc]] Gamma0[Nc, 
        2, 2]/(2 Beta0[4, Nc])],
                If[ms <= mu < mc, 
     MatrixExp[
      Log[
        AlphaS[mu, Lambda, 3, Nc]/AlphaS[m, Lambda, 3, Nc]] Gamma0[Nc,
          1, 2]/(2 Beta0[3, Nc])],
                If[md <= mu < ms, 
      MatrixExp[
       Log[
         AlphaS[mu, Lambda, 2, Nc]/AlphaS[m, Lambda, 2, Nc]] Gamma0[
          Nc, 1, 1]/(2 Beta0[2, Nc])], 0]]]]]

are the definition of $U^{(0)}$ which has different values depending on $\mu$. I did not use $V$ for the evaluation of $U^{(0)}$ since I think that if I use the non-diagonal form of $\gamma^{(0)}$ is should not be necessary. But still, $V$ is needed for some further calculations.

Where my problem lies is in the following

Is the implementation for the $V$ matrix correct?
Why when I try to output the diagonal form of $\gamma^{(0)}$ with parameters, let's say $N_c=3, N_u=1, N_d=0$ I get the following result?

Gamma0[Nc, 1, 0]
DiagonalizableMatrixQ[Gamma0[Nc, 1, 0]]
Gamma0D[Nc, 1, 0]
{{-(14/3), 6, 0, 0, 0, 0, 16/9, 0, 16/9, 0}, {6, -(14/3), -(2/9), 2/
  3, -(2/9), 2/3, 16/27, 0, 16/27, 0}, {0, 0, -(22/9), 22/3, -(4/9), 
  4/3, 32/27, 0, -(40/27), 0}, {0, 0, 52/9, -(4/3), -(2/9), 2/
  3, -(32/27), 0, -(32/27), -(8/3)}, {0, 0, 0, 0, 2, -6, 40/9, 0, 16/
  9, 0}, {0, 0, -(2/9), 2/3, -(2/9), -(142/3), 16/27, 8/3, 16/27, 
  0}, {0, 0, 0, 0, 4/3, 0, 46/9, -6, 16/9, 0}, {0, 0, -(2/9), 2/
  3, -(2/9), 2, 16/27, -(44/3), 16/27, 0}, {0, 0, -(10/9), -(2/3), 2/
  9, -(2/3), 56/27, 0, -(34/27), 6}, {0, 0, -(2/9), -(2/3), -(2/9), 2/
  3, 40/27, 0, 202/27, -(10/3)}}
True
{{Root[183264739328 - 368304734208 # + 118605081600 #^2 + 
   15910929792 #^3 - 4571121600 #^4 - 395263800 #^5 + 37995480 #^6 + 
   3735396 #^7 + 59049 #^8& , 1, 0], 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 
  Root[183264739328 - 368304734208 # + 118605081600 #^2 + 
   15910929792 #^3 - 4571121600 #^4 - 395263800 #^5 + 37995480 #^6 + 
   3735396 #^7 + 59049 #^8& , 2, 0], 0, 0, 0, 0, 0, 0, 0, 0}, {0, 
  0, -(32/3), 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, Root[
  183264739328 - 368304734208 # + 118605081600 #^2 + 15910929792 #^3 - 
   4571121600 #^4 - 395263800 #^5 + 37995480 #^6 + 3735396 #^7 + 
   59049 #^8& , 3, 0], 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, Root[
  183264739328 - 368304734208 # + 118605081600 #^2 + 15910929792 #^3 - 
   4571121600 #^4 - 395263800 #^5 + 37995480 #^6 + 3735396 #^7 + 
   59049 #^8& , 8, 0], 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, Root[
  183264739328 - 368304734208 # + 118605081600 #^2 + 15910929792 #^3 - 
   4571121600 #^4 - 395263800 #^5 + 37995480 #^6 + 3735396 #^7 + 
   59049 #^8& , 4, 0], 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, Root[
  183264739328 - 368304734208 # + 118605081600 #^2 + 15910929792 #^3 - 
   4571121600 #^4 - 395263800 #^5 + 37995480 #^6 + 3735396 #^7 + 
   59049 #^8& , 7, 0], 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, Root[
  183264739328 - 368304734208 # + 118605081600 #^2 + 15910929792 #^3 - 
   4571121600 #^4 - 395263800 #^5 + 37995480 #^6 + 3735396 #^7 + 
   59049 #^8& , 6, 0], 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 4/3, 0}, {0, 0,
   0, 0, 0, 0, 0, 0, 0, Root[
  183264739328 - 368304734208 # + 118605081600 #^2 + 15910929792 #^3 - 
   4571121600 #^4 - 395263800 #^5 + 37995480 #^6 + 3735396 #^7 + 
   59049 #^8& , 5, 0]}}

I do not now what is that Root and what is doing!

If the linked question does not sufficiently resolve your confusion, edit this question to explain why, and it can be reopened. — J. M.'s missing motivation, Jun 26 '21 at 15:11

On the diagonalization of matrices and strange results

0 Answers0