diagonalizeViaCongruence -- diagonalizes a symmetric matrix via congruence

• Usage:

  diagonalizeViaCongruence M

• Inputs:
  • M, a matrix, a symmetric matrix over any field or finite étale algebras over a field
• Outputs:
  • a matrix, a diagonal matrix congruent to M

i1 : R = QQ[x]/(x^2 - 1)

o1 = R

o1 : QuotientRing

i2 : M = matrix(R, {{1,2},{2,x}});

             2      2
o2 : Matrix R  <-- R

i3 : diagonalizeViaCongruence M

o3 = | 1 0   |
     | 0 x-4 |

             2      2
o3 : Matrix R  <-- R