Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
the eigenvalue.
If the option extend is set to False, then only the eigenvalues that live in the base ring are considered.
EXAMPLES: We compute the left eigenvectors of a 3 × 3 rational matrix.
sage: A = matrix(QQ,3,3,range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: es = A.eigenvectors_left(); es
[(0, [
(1, -2, 1)
], 1),
(-1.348469228349535, [(1, 0.3101020514433644, -0.3797958971132713)], 1),
(13.34846922834954, [(1, 1.289897948556636, 1.579795897113272)], 1)]
sage: eval, [evec], mult = es[0]
sage: delta = eval*evec - evec*A
sage: abs(abs(delta)) < 1e-10
True
Notice the difference between considering ring extensions or not.
sage: M=matrix(QQ,[[0,-1,0],[1,0,0],[0,0,2]])
sage: M.eigenvectors_left()
[(2, [
(0, 0, 1)
], 1), (-1*I, [(1, -1*I, 0)], 1), (1*I, [(1, 1*I, 0)], 1)]
sage: M.eigenvectors_left(extend=False)
[(2, [
(0, 0, 1)
], 1)]
left_kernel(*args, **kwds)
Returns the left kernel of this matrix, as a vector space or free module. This is the set of vectors x such
that x*self = 0.
Note: For the right kernel, use right_kernel(). The method kernel() is exactly equal to
left_kernel().
INPUT:
•algorithm - default: ‘default’ - a keyword that selects the algorithm employed. Allowable values
are:
–‘default’ - allows the algorithm to be chosen automatically
–‘generic’ - naive algorithm usable for matrices over any field
–‘pari’ - PARI library code for matrices over number fields or the integers
–‘padic’ - padic algorithm from IML library for matrices over the rationals and integers
–‘pluq’ - PLUQ matrix factorization for matrices mod 2
•basis - default: ‘echelon’ - a keyword that describes the format of the basis used to construct the
left kernel. Allowable values are:
–‘echelon’: the basis matrix is in echelon form
210 Chapter 7. Base class for matrices, part 2