Hi, there: I am computing the eigenvalues and eigenvectors of the following complex matrix. It looks that Sage does not give the right answer?

savecancel

The ring of complex floats CDF is not an exact ring:

sage: CDF.is_exact()
False

So while doing computation in floating point entries (i.e. in RDF or CDF) some errors may occur (see http://doc.sagemath.org/html/en/const...). You can change the base ring to the algebraic field QQbar:

sage: B = matrix(QQbar, [ [-1, 0, 1, 0, 0, 0, 1, I, -1 ], [0, 0, 0, -I, 1, I, 1, I, 0], [-1, -I, 1, -I, 1, 0, 1, 0, I], [0, 0, 0, 0, 1, I, 1, I, -1], [0, -I, 1, -I, 1, I, 1, I, 0], [-1, -I, 1, -I, 1, 0, 0, 0, 0], [-1, 0, 1, 0, 1, I, 1, I, -1], [0, -I, 1, -I, 1, I, 0, 0, 0], [-1, -I, 1, 0, 0, 0, 1, 0, -1]])
sage: B.eigenvalues()
[3.917412364954442? + 0.05429188890228639?*I, 0.6516886324409428? + 0.2353750083092179?*I, 0.3580569417083658? - 0.08398467888210074?*I, -0.1167390153623098? + 0.6697900534213050?*I, -0.1742956154446435? - 0.5989951966469979?*I, -0.7276189211835305? + 0.3726506970413855?*I, -0.8274431770819343? + 1.205961277873422?*I, -0.9961206931117568? - 1.191900422440107?*I, -1.084940516919576? - 0.6631886275784114?*I]

An alternative is to use an algebraic extension of QQusing NumberField:

sage: K.<a> = NumberField(x^2 + 1)
sage: K.is_exact()
True
sage: B = matrix(K, [ [-1, 0, 1, 0, 0, 0, 1, a, -1 ], [0, 0, 0, -a, 1, a, 1, a, 0], [-1, -a, 1, -a, 1, 0, 1, 0, a], [0, 0, 0, 0, 1, a, 1, a, -1], [0, -a, 1, -a, 1, a, 1, a, 0], [-1,-a, 1, -a, 1, 0, 0, 0, 0], [-1, 0, 1, 0, 1, a, 1, a, -1], [0, -a, 1, -a, 1, a, 0, 0, 0], [-1, -a, 1, 0, 0, 0, 1, 0, -1]])
sage: B.eigenvalues()
[-1.084940516919576? - 0.6631886275784114?*I, -0.9961206931117568? - 1.191900422440107?*I, -0.8274431770819343? + 1.205961277873422?*I, -0.7276189211835305? + 0.3726506970413855?*I, -0.1742956154446435? - 0.5989951966469979?*I, -0.1167390153623098? + 0.6697900534213050?*I, 0.3580569417083658? - 0.08398467888210074?*I, 0.6516886324409428? + 0.2353750083092179?*I, 3.917412364954442? + 0.05429188890228639?*I]