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One can go directly for the corresponding kernel. For instance:

sage: A = matrix( QQ, 3,3, [4,1,1, 1,4,1, 1,1,4] )
sage: A
[4 1 1]
[1 4 1]
[1 1 4]

sage: E = matrix.identity( A.nrows() )
sage: A.eigenvalues()
[6, 3, 3]

sage: ( A - 3*E ).kernel().basis()
[
(1, 0, -1),
(0, 1, -1)
]