Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: k = t-39
sage: k.restrict(V)
[ 0 -10 -12]
[ 0 -37 -1]
[ 0 2 -41]
sage: ker = k.kernel_on(V); ker
Vector space of degree 4 and dimension 1 over Rational Field
Basis matrix:
[ 1 -2/7 0 -2/7]
sage: ker.0 * k
(0, 0, 0, 0)
Test that trac ticket #9425 is fixed.
sage: V = span([[1/7,0,0] ,[0,1,0]], ZZ); V
Free module of degree 3 and rank 2 over Integer Ring
Echelon basis matrix:
[1/7 0 0]
[ 0 1 0]
sage: T = matrix(ZZ,3,[1,0,0,0,0,0,0,0,0]); T
[1 0 0]
[0 0 0]
[0 0 0]
sage: W = T.kernel_on(V); W.basis()
[
(0, 1, 0)
]
sage: W.is_submodule(V)
True
left_eigenmatrix()
Return matrices D and P, where D is a diagonal matrix of eigenvalues and P is the corresponding matrix
where the rows are corresponding eigenvectors (or zero vectors) so that P*self = D*P.
EXAMPLES:
sage: A = matrix(QQ,3,3,range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: D, P = A.eigenmatrix_left()
sage: D
[ 0 0 0]
[ 0 -1.348469228349535 0]
[ 0 0 13.34846922834954]
sage: P
[ 1 -2 1]
[ 1 0.3101020514433644 -0.3797958971132713]
[ 1 1.289897948556636 1.579795897113272]
sage: P*A == D*P
True
Because P is invertible, A is diagonalizable.
sage: A == (~P)*D*P
True
The matrix P may contain zero rows corresponding to eigenvalues for which the algebraic multiplicity is
greater than the geometric multiplicity. In these cases, the matrix is not diagonalizable.
204 Chapter 7. Base class for matrices, part 2