test failure in matrix/special.py

[collares]

Steps To Reproduce

sage -t --long --random-seed=214483688537001353062577859282581597229 /nix/store/0yv879ir56m3zyd95hdfw0yr4vznrk75-sage-src-10.2/src/sage/matrix/special.py
**********************************************************************
File "/nix/store/0yv879ir56m3zyd95hdfw0yr4vznrk75-sage-src-10.2/src/sage/matrix/special.py", line 3107, in sage.matrix.special.random_diagonalizable_matrix
Failed example:
    all(x in ZZ for x in (B-(6*identity_matrix(6))).rref().list())
Expected:
    True
Got:
    False

Environment

- **OS**: Linux (NixOS 23.11)
- **Sage Version**: 10.2

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide

[fchapoton]

Here is the problematic matrix, as a reproducible failing test:

B = matrix(QQ, [[-12, -36, 12, 24, -84, 108], [18, -12, 18, 0, 72, 36], [18, 0, -6, -24, 12, 0], [-18, -24, 12, 30, -60, 72], [0, 6, 0, 0, 18, -18], [6, 0, 6, 0, 36, 0]])
[x for x in (B-(6*identity_matrix(6))).rref().list() if x not in ZZ]

[tornaria]

A simpler reproducer that may shed some more light:

sage: set_random_seed(42)
sage: B = sage.matrix.special.random_diagonalizable_matrix(MatrixSpace(QQ,4),eigenvalues=[0,0,1],dimensions=[1,1,2])
sage: B.rref()
[   1    0  2/3  1/3]
[   0    1 -5/3 -4/3]
[   0    0    0    0]
[   0    0    0    0]

The doctest claims that all those should be integers. This fails not always but quite often (in the doctest, the three eigenvalues are taken at random, and we only loose when there are two eigenvalues that are both 4, both 6, or both -12, that's much less likely to happen).

[tornaria]

Original test (92837ca):

        sage: B=random_matrix(QQ, 6, algorithm='diagonalizable', eigenvalues=[-12,4,6],dimensions=[2,3,1]); B # random
        [ -52   32  240 -464  -96 -520]
        [   6    4  -48   72   36   90]
        [  46  -32 -108  296  -12  274]
        [  24  -16  -64  164    0  152]
        [  18  -16    0   72  -48   30]
        [   2    0  -16   24   12   34]
        sage: all([x in ZZ for x in (B-(-12*identity_matrix(6))).rref().list()])
        True
        sage: all([x in ZZ for x in (B-(4*identity_matrix(6))).rref().list()])
        True
        sage: all([x in ZZ for x in (B-(6*identity_matrix(6))).rref().list()])
        True
        sage: S=B.right_eigenmatrix()[1]; S_inverse=S.inverse(); S_inverse*B*S # random
        [  6   0   0   0   0   0]
        [  0 -12   0   0   0   0]
        [  0   0 -12   0   0   0]
        [  0   0   0   4   0   0]
        [  0   0   0   0   4   0]
        [  0   0   0   0   0   4]

Current test (c1c315c)

        sage: eigenvalues = [ZZ.random_element() for _ in range(3)]
        sage: B = random_matrix(QQ, 6, algorithm='diagonalizable',
        ....:                   eigenvalues=eigenvalues, dimensions=[2,3,1])
        sage: all(x in ZZ for x in (B-(-12*identity_matrix(6))).rref().list())
        True
        sage: all(x in ZZ for x in (B-(4*identity_matrix(6))).rref().list())
        True
        sage: all(x in ZZ for x in (B-(6*identity_matrix(6))).rref().list())
        True

I'm not sure this makes sense since the .rref() will be the identity most of the time. We really want to subtract the eigenvalues here, don't we?

Still there seems to be a bug when the eigenvalues argument has repeated values. Either it should raise a value error, or return a matrix which has integral rref.