# Revision history [back]

I am not sure about your exact question. Does the following help ?

sage: m = matrix([[6,-2,-1],[3,1,-1],[2,-1,2]])
sage: m.eigenvalues()
[3, 3, 3]
sage: m.eigenvectors_right()
[(3, [
(1, 1, 1)
], 3)]
sage: a = m-3
sage: a
[ 3 -2 -1]
[ 3 -2 -1]
[ 2 -1 -1]
sage: a.right_kernel()
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[1 1 1]
sage: (a^2).right_kernel()
Free module of degree 3 and rank 2 over Integer Ring
Echelon basis matrix:
[1 1 0]
[0 0 1]
sage: (a^3).right_kernel()
Free module of degree 3 and rank 3 over Integer Ring
Echelon basis matrix:
[1 0 0]
[0 1 0]
[0 0 1]


I am not sure about your exact question. Does the following help ?

sage: m = matrix([[6,-2,-1],[3,1,-1],[2,-1,2]])
sage: m.eigenvalues()
[3, 3, 3]
sage: m.eigenvectors_right()
[(3, [
(1, 1, 1)
], 3)]
sage: a = m-3
sage: a
[ 3 -2 -1]
[ 3 -2 -1]
[ 2 -1 -1]
sage: a.right_kernel()
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[1 1 1]
sage: (a^2).right_kernel()
Free module of degree 3 and rank 2 over Integer Ring
Echelon basis matrix:
[1 1 0]
[0 0 1]
sage: (a^2).right_kernel().basis()
[
(1, 1, 0),
(0, 0, 1)
]
sage: (a^3).right_kernel()
Free module of degree 3 and rank 3 over Integer Ring
Echelon basis matrix:
[1 0 0]
[0 1 0]
[0 0 1]
sage: (a^3).right_kernel().basis()
[
(1, 0, 0),
(0, 1, 0),
(0, 0, 1)
]