Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
... [3*a + 5, a + 4, 5*a + 6]])
sage: A.is_similar(B)
True
sage: B.is_similar(A)
True
sage: _, T = A.is_similar(B, transformation=True)
sage: T
[ 1 0 0]
[6*a + 1 4*a + 3 4*a + 2]
[6*a + 3 3*a + 5 3*a + 6]
sage: A == T.inverse()*B*T
True
Two matrices with different sets of eigenvalues, so they cannot possibly be similar.
sage: A = matrix(QQ, [[ 2, 3, -3, -6],
... [ 0, 1, -2, -8],
... [-3, -3, 4, 3],
... [-1, -2, 2, 6]])
sage: B = matrix(QQ, [[ 1, 1, 2, 4],
... [-1, 2, -3, -7],
... [-2, 3, -4, -7],
... [ 0, -1, 0, 0]])
sage: A.eigenvalues() == B.eigenvalues()
False
sage: A.is_similar(B, transformation=True)
(False, None)
Similarity is an equivalence relation, so this routine computes a representative of the equivalence class
for each matrix, the Jordan form, as provided by jordan_form(). The matrices below have identical
eigenvalues (as evidenced by equal characteristic polynomials), but slightly different Jordan forms, and
hence are not similar.
sage: A = matrix(QQ, [[ 19, -7, -29],
... [-16, 11, 30],
... [ 15, -7, -25]])
sage: B = matrix(QQ, [[-38, -63, 42],
... [ 14, 25, -14],
... [-14, -21, 18]])
sage: A.charpoly() == B.charpoly()
True
sage: A.jordan_form()
[-3| 0 0]
[--+-----]
[ 0| 4 1]
[ 0| 0 4]
sage: B.jordan_form()
[-3| 0| 0]
[--+--+--]
[ 0| 4| 0]
[--+--+--]
[ 0| 0| 4]
sage: A.is_similar(B)
False
Obtaining the Jordan form requires computing the eigenvalues of the matrix, which may not lie in the
field used for entries of the matrix. So the routine first checks the characteristic polynomials - if they are
unequal, then the matrices cannot be similar. However, when the characteristic polynomials are equal, we
must examine the Jordan form. In this case, the method may fail, EVEN when the matrices are similar.
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