Sage produces non-invertible transformation matrix to SR Jordan form
Consider the following matrix:
M = 1/4 * matrix([(0, 0, -(I + 1)*sqrt(2), -(I + 1)*sqrt(2), (I + 1)*sqrt(2), (I + 1)*sqrt(2)),
(0, 0, -(I - 1)*sqrt(2), (I - 1)*sqrt(2), (I - 1)*sqrt(2), -(I - 1)*sqrt(2)),
((I - 1)*sqrt(2), (I - 1)*sqrt(2), I - 1, I - 1, I - 1, I - 1),
(-(I + 1)*sqrt(2), (I + 1)*sqrt(2), -I - 1, I + 1, -I - 1, I + 1),
((I - 1)*sqrt(2), (I - 1)*sqrt(2), -I + 1, -I + 1, -I + 1, -I + 1),
(-(I + 1)*sqrt(2), (I + 1)*sqrt(2), I + 1, -I - 1, I + 1, -I - 1)])
The matrix M is diagonalizable. Indeed, let z = exp(i*pi/3); then M is similar to J = diagonal_matrix([ -1, -1, z, z, z.conjugate(), z.conjugate() ]). For example, one can use matrix(QQbar, M).is_similar(matrix(QQbar, J)) to obtain a similarity transformation. We must embed M (implicitly defined over SR) into QQbar to have exactness to perform the similarity computation. Indeed, both M.is_similar(J) and M.diagonalization() raise respectively TypeError and ValueError as expected. Since M is diagonalizable, the diagonalization agrees (up to permutation) with the Jordan form matrix(QQbar, M).jordan_form() as expected.
In SageMath version 10.4.beta3 as well as 9.5 I can also run M.jordan_form() with no errors thrown; the answer agrees with matrix(QQbar, M).jordan_form(). However, matrix(QQbar, M).jordan_form(transformation=True) and M.jordan_form(transformation=True) do not agree; the latter produces a transformation matrix that is not invertible!
The documentation for Matrix_symbolic_dense.jordan_form suggests that matrices over SR support Jordan form with the transformation flag set to true--why in this case does Sage appear to produce a non-invertible transformation matrix for a correctly computed Jordan form? Or should I in general be converting away from SR to perform similarity transformations in Sage?
Looks like a bug. Try to find a smaller instance if you can.
This particular example comes from the representation of the double cover of the alternating group A5 in SU(2). It is a tensor product of
matrix([(0, -sqrt(2), sqrt(2)), (I*sqrt(2), I, I), (I*sqrt(2), -I, -I)])andmatrix([(I + 1, I + 1), (I - 1, -I + 1)]), thoughjordan_form(over SR) computes transformation matrices correctly for both tensor factors.Upon further digging the bug appears to be in Maxima, not Sage. The erroneous transformation matrix comes from
M._maxima_lib_().diag_mode_matrix(M._maxima_().jordan()).Try to reproduce it directly in maxima, and then file a bug in maxima bug-tracker.