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Eigensystem without Root objects

Mathematica Asked on June 15, 2021

I’d like to find eigenvalues and eigenvectors of the parametric, 21×21, matrix below without Root objects.

My matrix is:

{{-100 d, 0, 2 Sqrt[190] e, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
 0, 0, 0, 0}, {0, -81 d, 0, 6 Sqrt[57] e, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0}, {2 Sqrt[190] e, 0, -64 d, 0, 6 Sqrt[102] e,
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 6 Sqrt[57] e, 
 0, -49 d, 0, 8 Sqrt[85] e, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 0},{0, 0, 6 Sqrt[102] e, 0, -36 d, 0, 60 Sqrt[2] e, 0, 0, 0, 0, 0,
 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 8 Sqrt[85] e, 0, -25 d, 0, 
 42 Sqrt[5] e, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 
 60 Sqrt[2] e, 0, -16 d, 0, 28 Sqrt[13] e, 0, 0, 0, 0, 0, 0, 0, 0, 0,
 0, 0, 0}, {0, 0, 0, 0, 0, 42 Sqrt[5] e, 0, -9 d, 0, 104 e, 0, 0, 0,
 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 28 Sqrt[13] e, 0, -4 d,
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 
 12 Sqrt[78] e, 0, -d, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0,
 0, 0, 0, 0, 6 Sqrt[330] e, 0, 0, 0, 6 Sqrt[330] e, 0, 0, 0, 0, 0, 
 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -d, 0, 12 Sqrt[78] e, 0,
 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4 d, 0, 
 28 Sqrt[13] e, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
 12 Sqrt[78] e, 0, -9 d, 0, 42 Sqrt[5] e, 0, 0, 0, 0, 0}, {0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 28 Sqrt[13] e, 0, -16 d, 0, 60 Sqrt[2] e,
 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42 Sqrt[5] e, 
 0, -25 d, 0, 8 Sqrt[85] e, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
 0, 0, 0, 0, 60 Sqrt[2] e, 0, -36 d, 0, 6 Sqrt[102] e, 0, 0}, {0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8 Sqrt[85] e, 0, -49 d, 0, 
 6 Sqrt[57] e, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
 6 Sqrt[102] e, 0, -64 d, 0, 2 Sqrt[190] e}, {0, 0, 0, 0, 0, 0, 0, 0,
 0, 0, 0, 0, 0, 0, 0, 0, 0, 6 Sqrt[57] e, 0, -81 d, 0}, {0, 0, 0, 0,
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2 Sqrt[190] e, 
 0, -100 d}}
linear algebramatrixsymbolic

One Answer

I confirm

Det[mat]=0
Dimensions[mat]=21

The problem of mat is the column 11. There are only 0 in this column.

The matrix gets easy for e==0:

{evals, evects} = mat /. e -> 0 // Eigensystem

{{-100 d, -100 d, -81 d, -81 d, -64 d, -64 d, -49 d, -49 d, -36 d, 
-36 d, -25 d, -25 d, -16 d, -16 d, -9 d, -9 d, -4 d, -4 d, -d, -d, 
  0}, {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 
   0}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 
   0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 
   0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 
   0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}}}

and d==0:

{evals, evects} = mat /. d -> 0 // Eigensystem


{{-2 Sqrt[3443 + 10 Sqrt[54637]] e, 
  2 Sqrt[3443 + 10 Sqrt[54637]] e, -2 Sqrt[
   2039 + 156 Sqrt[78] + 2 Sqrt[1231141 + 12948 Sqrt[78]]] e, 
  2 Sqrt[2039 + 156 Sqrt[78] + 2 Sqrt[1231141 + 12948 Sqrt[78]]]
    e, -4 Sqrt[2 (341 + 45 Sqrt[33])] e, -4 Sqrt[
   2 (341 + 45 Sqrt[33])] e, 4 Sqrt[2 (341 + 45 Sqrt[33])] e, 
  4 Sqrt[2 (341 + 45 Sqrt[33])] e, -2 Sqrt[3443 - 10 Sqrt[54637]] e, 
  2 Sqrt[3443 - 10 Sqrt[54637]] e, -2 Sqrt[
   2039 + 156 Sqrt[78] - 2 Sqrt[1231141 + 12948 Sqrt[78]]] e, 
  2 Sqrt[2039 + 156 Sqrt[78] - 2 Sqrt[1231141 + 12948 Sqrt[78]]]
    e, -4 Sqrt[2 (341 - 45 Sqrt[33])] e, -4 Sqrt[
   2 (341 - 45 Sqrt[33])] e, 4 Sqrt[2 (341 - 45 Sqrt[33])] e, 
  4 Sqrt[2 (341 - 45 Sqrt[33])] e, 0, 0, 0, 0, 
  0}, {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (
   157 Sqrt[8398] + Sqrt[458841526])/6783, 
   0, (-157 Sqrt[969 (3443 + 10 Sqrt[54637])] - Sqrt[
    52943253 (3443 + 10 Sqrt[54637])])/122094, 0, (
   293 Sqrt[4845] + Sqrt[264716265])/5814, 
   0, -(1/3) Sqrt[1/57 (3443 + 10 Sqrt[54637])], 0, 1, 0}, {0, 0, 0, 
   0, 0, 0, 0, 0, 0, 0, 0, (157 Sqrt[8398] + Sqrt[458841526])/6783, 
   0, (157 Sqrt[969 (3443 + 10 Sqrt[54637])] + Sqrt[
    52943253 (3443 + 10 Sqrt[54637])])/122094, 0, (
   293 Sqrt[4845] + Sqrt[264716265])/5814, 0, 
   1/3 Sqrt[1/57 (3443 + 10 Sqrt[54637])], 0, 1, 
   0}, {0, (-2028 Sqrt[969] - 83 Sqrt[8398] + Sqrt[
    8398 (1231141 + 12948 Sqrt[78])])/185640, 0, (1/
   1670760)(6084 Sqrt[
      17 (2039 + 156 Sqrt[78] + 2 Sqrt[1231141 + 12948 Sqrt[78]])] + 
     83 Sqrt[1326 (2039 + 156 Sqrt[78] + 
         2 Sqrt[1231141 + 12948 Sqrt[78]])] - Sqrt[
     1326 (1231141 + 12948 Sqrt[78]) (2039 + 156 Sqrt[78] + 
        2 Sqrt[1231141 + 12948 Sqrt[78]])]), 
   0, (-12168 Sqrt[5] + 2039 Sqrt[390] + 
    2 Sqrt[390 (1231141 + 12948 Sqrt[78])])/49140, 
   0, -(1/6) Sqrt[
    1/78 (2039 + 156 Sqrt[78] + 2 Sqrt[1231141 + 12948 Sqrt[78]])], 0,
    1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, (-2028 Sqrt[969] - 83 Sqrt[8398] + Sqrt[
    8398 (1231141 + 12948 Sqrt[78])])/185640, 
   0, -(13/105) Sqrt[
     1/17 (2039 + 156 Sqrt[78] + 
        2 Sqrt[1231141 + 12948 Sqrt[78]])] - (1/21420)
    Sqrt[1/78 (2039 + 156 Sqrt[78] + 
        2 Sqrt[1231141 + 12948 Sqrt[78]])] (83 Sqrt[17] - 
       78 Sqrt[1326] - Sqrt[17 (1231141 + 12948 Sqrt[78])]), 
   0, (-12168 Sqrt[5] + 2039 Sqrt[390] + 
    2 Sqrt[390 (1231141 + 12948 Sqrt[78])])/49140, 0, 
   1/6 Sqrt[
    1/78 (2039 + 156 Sqrt[78] + 2 Sqrt[1231141 + 12948 Sqrt[78]])], 0,
    1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {-1, 0, 
   2 Sqrt[1/95 (341 + 45 Sqrt[33])], 
   0, -((3 (47 Sqrt[4845] + 20 Sqrt[17765]))/1615), 0, (
   2 (3 Sqrt[4845 (341 + 45 Sqrt[33])] + 
      2 Sqrt[17765 (341 + 45 Sqrt[33])]))/1615, 
   0, -((7 (3 Sqrt[125970] + 2 Sqrt[461890]))/1615), 0, 0, 0, (
   7 (3 Sqrt[125970] + 2 Sqrt[461890]))/1615, 
   0, -((2 (3 Sqrt[4845 (341 + 45 Sqrt[33])] + 
       2 Sqrt[17765 (341 + 45 Sqrt[33])]))/1615), 0, (
   3 (47 Sqrt[4845] + 20 Sqrt[17765]))/1615, 
   0, -2 Sqrt[1/95 (341 + 45 Sqrt[33])], 0, 1}, {(
   Sqrt[1/165 (341 + 45 Sqrt[33])] (3 Sqrt[125970] - 2 Sqrt[461890]))/
   4641, 0, (2 (277 Sqrt[6630] - 9 Sqrt[24310]))/69615, 
   0, -(1/91) Sqrt[
    1/165 (341 + 45 Sqrt[33])] (Sqrt[26] + 2 Sqrt[858]), 0, (
   2 (135 Sqrt[130] + 31 Sqrt[4290]))/4095, 
   0, -(2/3) Sqrt[1/165 (341 + 45 Sqrt[33])], 0, 1, 0, 0, 0, 0, 0, 0, 
   0, 0, 0, 0}, {-1, 0, -2 Sqrt[1/95 (341 + 45 Sqrt[33])], 
   0, -((3 (47 Sqrt[4845] + 20 Sqrt[17765]))/1615), 
   0, -((2 (3 Sqrt[4845 (341 + 45 Sqrt[33])] + 
       2 Sqrt[17765 (341 + 45 Sqrt[33])]))/1615), 
   0, -((7 (3 Sqrt[125970] + 2 Sqrt[461890]))/1615), 0, 0, 0, (
   7 (3 Sqrt[125970] + 2 Sqrt[461890]))/1615, 0, (
   2 (3 Sqrt[4845 (341 + 45 Sqrt[33])] + 
      2 Sqrt[17765 (341 + 45 Sqrt[33])]))/1615, 0, (
   3 (47 Sqrt[4845] + 20 Sqrt[17765]))/1615, 0, 
   2 Sqrt[1/95 (341 + 45 Sqrt[33])], 0, 
   1}, {-((Sqrt[
     1/165 (341 + 45 Sqrt[33])] (3 Sqrt[125970] - 2 Sqrt[461890]))/
    4641), 0, (2 (277 Sqrt[6630] - 9 Sqrt[24310]))/69615, 0, 
   1/91 Sqrt[1/165 (341 + 45 Sqrt[33])] (Sqrt[26] + 2 Sqrt[858]), 0, (
   2 (135 Sqrt[130] + 31 Sqrt[4290]))/4095, 0, 
   2/3 Sqrt[1/165 (341 + 45 Sqrt[33])], 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 
   0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (
   157 Sqrt[8398] - Sqrt[458841526])/6783, 
   0, (-157 Sqrt[969 (3443 - 10 Sqrt[54637])] + Sqrt[
    52943253 (3443 - 10 Sqrt[54637])])/122094, 0, (
   293 Sqrt[4845] - Sqrt[264716265])/5814, 
   0, -(1/3) Sqrt[1/57 (3443 - 10 Sqrt[54637])], 0, 1, 0}, {0, 0, 0, 
   0, 0, 0, 0, 0, 0, 0, 0, (157 Sqrt[8398] - Sqrt[458841526])/6783, 
   0, (157 Sqrt[969 (3443 - 10 Sqrt[54637])] - Sqrt[
    52943253 (3443 - 10 Sqrt[54637])])/122094, 0, (
   293 Sqrt[4845] - Sqrt[264716265])/5814, 0, 
   1/3 Sqrt[1/57 (3443 - 10 Sqrt[54637])], 0, 1, 
   0}, {0, (-2028 Sqrt[969] - 83 Sqrt[8398] - Sqrt[
    8398 (1231141 + 12948 Sqrt[78])])/185640, 0, 
   13/105 Sqrt[
     1/17 (2039 + 156 Sqrt[78] - 
        2 Sqrt[1231141 + 12948 Sqrt[78]])] + (1/21420)
    Sqrt[1/78 (2039 + 156 Sqrt[78] - 
        2 Sqrt[1231141 + 12948 Sqrt[78]])] (83 Sqrt[17] - 
       78 Sqrt[1326] + Sqrt[17 (1231141 + 12948 Sqrt[78])]), 
   0, (-12168 Sqrt[5] + 2039 Sqrt[390] - 
    2 Sqrt[390 (1231141 + 12948 Sqrt[78])])/49140, 
   0, -(1/6) Sqrt[
    1/78 (2039 + 156 Sqrt[78] - 2 Sqrt[1231141 + 12948 Sqrt[78]])], 0,
    1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, (-2028 Sqrt[969] - 83 Sqrt[8398] - Sqrt[
    8398 (1231141 + 12948 Sqrt[78])])/185640, 
   0, -(13/105) Sqrt[
     1/17 (2039 + 156 Sqrt[78] - 
        2 Sqrt[1231141 + 12948 Sqrt[78]])] - (1/21420)
    Sqrt[1/78 (2039 + 156 Sqrt[78] - 
        2 Sqrt[1231141 + 12948 Sqrt[78]])] (83 Sqrt[17] - 
       78 Sqrt[1326] + Sqrt[17 (1231141 + 12948 Sqrt[78])]), 
   0, (-12168 Sqrt[5] + 2039 Sqrt[390] - 
    2 Sqrt[390 (1231141 + 12948 Sqrt[78])])/49140, 0, 
   1/6 Sqrt[
    1/78 (2039 + 156 Sqrt[78] - 2 Sqrt[1231141 + 12948 Sqrt[78]])], 0,
    1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {-1, 0, 
   2 Sqrt[1/95 (341 - 45 Sqrt[33])], 
   0, -((3 (47 Sqrt[4845] - 20 Sqrt[17765]))/1615), 0, (
   2 (3 Sqrt[4845 (341 - 45 Sqrt[33])] - 
      2 Sqrt[17765 (341 - 45 Sqrt[33])]))/1615, 
   0, -((7 (3 Sqrt[125970] - 2 Sqrt[461890]))/1615), 0, 0, 0, (
   7 (3 Sqrt[125970] - 2 Sqrt[461890]))/1615, 
   0, -((2 (3 Sqrt[4845 (341 - 45 Sqrt[33])] - 
       2 Sqrt[17765 (341 - 45 Sqrt[33])]))/1615), 0, (
   3 (47 Sqrt[4845] - 20 Sqrt[17765]))/1615, 
   0, -2 Sqrt[1/95 (341 - 45 Sqrt[33])], 0, 1}, {(
   Sqrt[1/165 (341 - 45 Sqrt[33])] (3 Sqrt[125970] + 2 Sqrt[461890]))/
   4641, 0, -((2 (277 Sqrt[6630] + 9 Sqrt[24310]))/69615), 0, 
   1/91 Sqrt[1/165 (341 - 45 Sqrt[33])] (-Sqrt[26] + 2 Sqrt[858]), 
   0, (2 (-135 Sqrt[130] + 31 Sqrt[4290]))/4095, 
   0, -(2/3) Sqrt[1/165 (341 - 45 Sqrt[33])], 0, 1, 0, 0, 0, 0, 0, 0, 
   0, 0, 0, 0}, {-1, 0, -2 Sqrt[1/95 (341 - 45 Sqrt[33])], 
   0, -((3 (47 Sqrt[4845] - 20 Sqrt[17765]))/1615), 
   0, -((2 (3 Sqrt[4845 (341 - 45 Sqrt[33])] - 
       2 Sqrt[17765 (341 - 45 Sqrt[33])]))/1615), 
   0, -((7 (3 Sqrt[125970] - 2 Sqrt[461890]))/1615), 0, 0, 0, (
   7 (3 Sqrt[125970] - 2 Sqrt[461890]))/1615, 0, (
   2 (3 Sqrt[4845 (341 - 45 Sqrt[33])] - 
      2 Sqrt[17765 (341 - 45 Sqrt[33])]))/1615, 0, (
   3 (47 Sqrt[4845] - 20 Sqrt[17765]))/1615, 0, 
   2 Sqrt[1/95 (341 - 45 Sqrt[33])], 0, 
   1}, {-((Sqrt[
     1/165 (341 - 45 Sqrt[33])] (3 Sqrt[125970] + 2 Sqrt[461890]))/
    4641), 0, -((2 (277 Sqrt[6630] + 9 Sqrt[24310]))/69615), 
   0, -(1/91) Sqrt[
    1/165 (341 - 45 Sqrt[33])] (-Sqrt[26] + 2 Sqrt[858]), 0, (
   2 (-135 Sqrt[130] + 31 Sqrt[4290]))/4095, 0, 
   2/3 Sqrt[1/165 (341 - 45 Sqrt[33])], 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 
   0, 0}, {-1, 0, 0, 0, Sqrt[95/51]/3, 0, 0, 
   0, -((5 Sqrt[190/663])/7), 0, 0, 0, (5 Sqrt[190/663])/7, 0, 0, 
   0, -(Sqrt[(95/51)]/3), 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0, (21 Sqrt[19/442])/8, 0, 0, 0, -((3 Sqrt[57/85])/4), 0, 0, 0, 1, 
   0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0}, {0, (208 Sqrt[17/57])/63, 0, 0, 0, -(52/(21 Sqrt[5])), 0, 0, 0,
    1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}}

These two solutions are already very large.

For e<>0 and d<>0 the solution is fast calculated and contains many Roots objects. Some easy some complicated.

As written in the Mathematica documentation for ToRadicals: "The result may not be equal to the Root object for some values of the parameter:".

A representative Roots object is

evals[[2]]

Root[893025 d^5 - 1153929240 d^3 e^2 + 
   5543568720 d e^4 + (1057221 d^4 - 122223816 d^2 e^2 + 
      102248784 e^4) #1 + (172810 d^3 - 
      3463240 d e^2) #1^2 + (8778 d^2 - 27544 e^2) #1^3 + 
   165 d #1^4 + #1^5 &, 1]

It has order 5 and is therefore not always solvable in Roots and ToRadicals. This example shows that setting after applying the Eigensystem built-in the values might not give the ones as setting values before.

The coefficients vary but the structure is at most the same. A big constant polynomial of e and d, all orders have coefficients up to four and the fifth order has 1. There are only 20 non-zero eigenvalues. The 21th is zero due to the column 11.

evals[[21]]

Root[14745600 d^5 - 2557808640 d^3 e^2 + 
   4043243520 d e^4 + (5395456 d^4 - 184380416 d^2 e^2 + 
      50642944 e^4) #1 + (489280 d^3 - 
      3731200 d e^2) #1^2 + (16368 d^2 - 21824 e^2) #1^3 + 
   220 d #1^4 + #1^5 &, 5]

Root and ToRadical prefer numerical parameters for simplification and representation.

Based on column 11 is might be sensible to solve in smaller matrixes. That is not easily achieved.

It is this strange structure:

mat

mat

d on the diagonal and e in the two bands on band aside from the diagonal that makes this representation interesting. But column 11 can be permutated out like this Eigensystem results indicate so that the triband structure is interrupted.

mat1={
 {-100 d, 0, 2 Sqrt[190] e, 0, 0, 0, 0, 0, 0, 0}, {0, -81 d, 0, 
  6 Sqrt[57] e, 0, 0, 0, 0, 0, 0}, {2 Sqrt[190] e, 0, -64 d, 0, 
  6 Sqrt[102] e, 0, 0, 0, 0, 0}, {0, 6 Sqrt[57] e, 0, -49 d, 0, 
  8 Sqrt[85] e, 0, 0, 0, 0}, {0, 0, 6 Sqrt[102] e, 0, -36 d, 0, 
  60 Sqrt[2] e, 0, 0, 0}, {0, 0, 0, 8 Sqrt[85] e, 0, -25 d, 0, 
  42 Sqrt[5] e, 0, 0}, {0, 0, 0, 0, 60 Sqrt[2] e, 0, -16 d, 0, 
  28 Sqrt[13] e, 0}, {0, 0, 0, 0, 0, 42 Sqrt[5] e, 0, -9 d, 0, 
  104 e}, {0, 0, 0, 0, 0, 0, 28 Sqrt[13] e, 0, -4 d, 0}, {0, 0, 0, 0, 
  0, 0, 0, 12 Sqrt[78] e, 0, -d}}

has

Det[mat1]

1152 (-38400 d^5 + 6660960 d^3 e^2 - 10529280 d e^4) (-297675 d^5 + 
   13144680 d^3 e^2 + 41277600 Sqrt[78] d^3 e^2 - 6032880 d e^4 - 
   204647040 Sqrt[78] d e^4)

mat1 has to be taken into account with 6 Sqrt[330] e from row 11. The roots objects are similar to the ones from the complete given matrix.

mat2={{-d, 0, 12 Sqrt[78] e, 0, 0, 0, 0, 0, 0, 0}, {0, -4 d, 0, 
  28 Sqrt[13] e, 0, 0, 0, 0, 0, 0}, {12 Sqrt[78] e, 0, -9 d, 0, 
  42 Sqrt[5] e, 0, 0, 0, 0, 0}, {0, 28 Sqrt[13] e, 0, -16 d, 0, 
  60 Sqrt[2] e, 0, 0, 0, 0}, {0, 0, 42 Sqrt[5] e, 0, -25 d, 0, 
  8 Sqrt[85] e, 0, 0, 0}, {0, 0, 0, 60 Sqrt[2] e, 0, -36 d, 0, 
  6 Sqrt[102] e, 0, 0}, {0, 0, 0, 0, 8 Sqrt[85] e, 0, -49 d, 0, 
  6 Sqrt[57] e, 0}, {0, 0, 0, 0, 0, 6 Sqrt[102] e, 0, -64 d, 0, 
  2 Sqrt[190] e}, {0, 0, 0, 0, 0, 0, 6 Sqrt[57] e, 0, -81 d, 0}, {0, 
  0, 0, 0, 0, 0, 0, 2 Sqrt[190] e, 0, -100 d}}

Has the same factor from row 11.

Det[mat2]

(* 3456 (-99225 d^5 + 128214360 d^3 e^2 - 615952080 d e^4) (-38400 d^5 + 6660960 d^3 e^2 - 10529280 d e^4)*)

The root objects are similar to the ones from the complete matrix and of order 5. The coefficients of the polynomials are of the same magnitude then the complete matrix. So the effort is not reduced.

Matrix operation work only on non-parameter values. e==0 shows only a diagonal matrix has to subtracted to work with spurless banded matrix and d==0 shows the complexity of this.

For special values all works fine for numbers:

mat4 = mat /. d -> 1 /. e -> 1;
{a, b} = Eigensystem@mat4

Eigensystem e==1&&d==1

Answered by Steffen Jaeschke on June 15, 2021

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