Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
•other - a matrix, which should be square, and of the same size as self, where the entries of the
matrix have a fraction field equal to that of self. Inexact rings are not supported.
•transformation - default: False - if True, the output will include the change-of-basis matrix.
See below for an exact description.
OUTPUT:
Two matrices, A and B are similar if there is an invertible matrix S such that A = S −1 BS. S can be
interpreted as a change-of-basis matrix if A and B are viewed as matrix representations of the same linear
transformation.
When transformation=False this method will return True if such a matrix S exists, otherwise it
will return False. When transformation=True the method returns a pair. The first part of the pair
is True or False depending on if the matrices are similar and the second part is the change-of-basis
matrix, or None should it not exist.
When the transformation matrix is requested, it will satisfy self = S.inverse()*other*S.
If the base rings for any of the matrices is the integers, the rationals, or the field of algebraic numbers
(QQbar), then the matrices are converted to have QQbar as their base ring prior to checking the equality
of the base rings.
It is possible for this routine to fail over most fields, even when the matrices are similar. However, since
the field of algebraic numbers is algebraically closed, the routine will always produce a result for matrices
with rational entries.
EXAMPLES:
The two matrices in this example were constructed to be similar. The computations happen in the field of
algebraic numbers, but we are able to convert the change-of-basis matrix back to the rationals (which may
not always be possible).
sage: A = matrix(ZZ, [[-5, 2, -11],
... [-6, 7, -42],
... [0, 1, -6]])
sage: B = matrix(ZZ, [[ 1, 12, 3],
... [-1, -6, -1],
... [ 0, 6, 1]])
sage: A.is_similar(B)
True
sage: _, T = A.is_similar(B, transformation=True)
sage: T
[ 1.0000000000000 + 0.e-13*I 0.e-13 + 0.e-13*I 0.e-13 + 0.e-13*I]
[-0.6666666666667 + 0.e-13*I 0.16666666666667 + 0.e-14*I -0.8333333333334 + 0.e-13*I]
[ 0.6666666666667 + 0.e-13*I 0.e-13 + 0.e-13*I -0.333333333334 + 0.e-13*I]
sage: T.change_ring(QQ)
[ 1 0 0]
[-2/3 1/6 -5/6]
[ 2/3 0 -1/3]
sage: A == T.inverse()*B*T
True
Other exact fields are supported.
sage: F. = FiniteField(7^2)
sage: A = matrix(F,[[2*a + 5, 6*a + 6, a + 3],
... [ a + 3, 2*a + 2, 4*a + 2],
... [2*a + 6, 5*a + 5, 3*a]])
sage: B = matrix(F,[[5*a + 5, 6*a + 4, a + 1],
... [ a + 5, 4*a + 3, 3*a + 3],
192 Chapter 7. Base class for matrices, part 2