Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
This is not the case for matrices over the integers, rationals or algebraic numbers, since the computations
are done in the algebraically closed field of algebraic numbers.
Here is an example where the similarity is obvious, but the routine fails to compute a result.
sage: F. = FiniteField(7^2)
sage: C = matrix(F,[[ a + 2, 5*a + 4],
... [6*a + 6, 6*a + 4]])
sage: S = matrix(ZZ, [[0, 1],
... [1, 0]])
sage: D = S.inverse()*C*S
sage: C.is_similar(D)
Traceback (most recent call last):
...
ValueError: unable to compute Jordan canonical form for a matrix
sage: C.jordan_form()
Traceback (most recent call last):
...
RuntimeError: Some eigenvalue does not exist in Finite Field in a of size 7^2.
Inexact rings and fields are also not supported.
sage: A = matrix(CDF, 2, 2, range(4))
sage: B = copy(A)
sage: A.is_similar(B)
Traceback (most recent call last):
...
ValueError: unable to compute Jordan canonical form for a matrix
Rectangular matrices and mismatched sizes return quickly.
sage: A = matrix(3, 2, range(6))
sage: B = copy(A)
sage: A.is_similar(B)
False
sage: A = matrix(2, 2, range(4))
sage: B = matrix(3, 3, range(9))
sage: A.is_similar(B, transformation=True)
(False, None)
If the fraction fields of the entries are unequal, it is an error, except in the case when the rationals gets
promoted to the algebraic numbers.
sage: A = matrix(ZZ, 2, 2, range(4))
sage: B = matrix(GF(2), 2, 2, range(4))
sage: A.is_similar(B, transformation=True)
Traceback (most recent call last):
...
TypeError: matrices need to have entries with identical fraction fields, not Algebraic Field
sage: A = matrix(ZZ, 2, 2, range(4))
sage: B = matrix(QQbar, 2, 2, range(4))
sage: A.is_similar(B)
True
Inputs are checked.
sage: A = matrix(ZZ, 2, 2, range(4))
sage: A.is_similar(’garbage’)
Traceback (most recent call last):
...
194 Chapter 7. Base class for matrices, part 2