Your circulant matrix has a characteristic polynomial of degree 11, which has (probably) no explicit solution by radicals ; Sage has herefore little chance of finding them:
sage: M=matrix.circulant([1,-1/2*I,0,0,0,0,0,0,0,0,1/2*I])
sage: M.parent()
Full MatrixSpace of 11 by 11 dense matrices over Symbolic Ring
sage: M.charpoly()
x^11 - 11*x^10 + 209/4*x^9 - 561/4*x^8 + 935/4*x^7 - 1001/4*x^6 + 11011/64*x^5 - 4719/64*x^4 + 4719/256*x^3 - 605/256*x^2 + 121/1024*x - 1/1024
sage: M.charpoly().parent()
Univariate Polynomial Ring in x over Symbolic Ring
However, one can get a designation of these in QQbar. Moving your matrix to QQbar allows you to get your solution:
sage: Mprime=M.change_ring(QQbar)
Now, the characteristic polynomial has its coefficients in QQbar:
sage: Mprime.charpoly().parent()
Univariate Polynomial Ring in x over Algebraic Field
And Sage can use specialied methods for polynomials over QQbar:
sage: EV=Mprime.eigenvectors_right()
sage: len(EV)
11
You have indeed 11 solutions, which are triples (eigenvalue, eigenvector, multiplicity).Let's look at one:
sage: EV[0]
(1.989821441880933?, [
(1.00000000000? + 0.?e-11*I, -0.14231483828? + 0.98982144188?*I, -0.95949297362? - 0.28173255684?*I, 0.41541501300? - 0.90963199536?*I, 0.84125353283? + 0.54064081746?*I, -0.65486073395? + 0.75574957436?*I, -0.65486073395? - 0.75574957436?*I, 0.84125353283? - 0.54064081746?*I, 0.41541501300? + 0.90963199536?*I, -0.95949297362? + 0.28173255684?*I, -0.14231483828? - 0.98982144188?*I)
], 1)
A similar solution applies to your second problem:
sage: M2=matrix.circulant([1,-1/2*I,0,0,0,0,1/2*I])
sage: EV2_bad=M2.eigenvectors_right()
sage: EV2_bad[5]
(1/12*sqrt(144*(7/1152*I*sqrt(3) - 7/3456)^(1/3) + 7/(7/1152*I*sqrt(3) - 7/3456)^(1/3) + 84) + 1,
[],
1)
sage: EV2=M2.change_ring(QQbar).eigenvectors_right()
sage: EV2[5]
(0.2181685175319702?, [
(1.000000000000000? + 0.?e-15*I, 0.623489801858734? - 0.781831482468030?*I, -0.222520933956314? - 0.974927912181824?*I, -0.900968867902419? - 0.433883739117558?*I, -0.900968867902419? + 0.433883739117558?*I, -0.222520933956315? + 0.974927912181824?*I, 0.623489801858734? + 0.781831482468030?*I)
], 1)
HTH,
