# 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?

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Looks like a bug. Try to find a smaller instance if you can.

( 2024-04-25 12:01:48 +0200 )edit

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)]) and matrix([(I + 1, I + 1), (I - 1, -I + 1)]), though jordan_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()).

( 2024-04-25 13:58:53 +0200 )edit

Try to reproduce it directly in maxima, and then file a bug in maxima bug-tracker.

( 2024-04-26 08:00:17 +0200 )edit