Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: b = a.change_ring(R.fraction_field())
sage: b.echelon_form()
# potentially useful
[ 1 y/x]
[ 0 0]
Echelon form is not defined over arbitrary rings:
sage: a = matrix(Integers(9),3,range(9))
sage: a.echelon_form()
Traceback (most recent call last):
...
NotImplementedError: Echelon form not implemented over ’Ring of integers modulo 9’.
Involving a sparse matrix:
sage: m = matrix(3,[1, 1, 1, 1, 0, 2, 1, 2, 0], sparse=True); m
[1 1 1]
[1 0 2]
[1 2 0]
sage: m.echelon_form()
[ 1 0 2]
[ 0 1 -1]
[ 0 0 0]
sage: m.echelonize(); m
[ 1 0 2]
[ 0 1 -1]
[ 0 0 0]
The transformation matrix is optionally returned:
sage: m_original = m
sage: transformation_matrix = m.echelonize(transformation=True)
sage: m == transformation_matrix * m_original
True
eigenmatrix_left()
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.
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