Elements of Finite Algebras

class sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element.FiniteDimensionalAlgebraElement(A, elt=None, check=True)

Bases: sage.structure.element.AlgebraElement

Create an element of a FiniteDimensionalAlgebra using a multiplication table.

INPUT:

  • A – a FiniteDimensionalAlgebra which will be the parent
  • elt – vector, matrix or element of the base field (default: None)
  • check – boolean (default: True); if False and elt is a matrix, assume that it is known to be the matrix of an element

If elt is a vector, it is interpreted as a vector of coordinates with respect to the given basis of A. If elt is a matrix, it is interpreted as a multiplication matrix with respect to this basis.

EXAMPLES:

sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1,0], [0,1]]), Matrix([[0,1], [0,0]])])
sage: A(17)
2*e0
sage: A([1,1])
e0 + e1
characteristic_polynomial()

Return the characteristic polynomial of self.

Note

This function just returns the characteristic polynomial of the matrix of right multiplication by self. This may not be a very meaningful invariant if the algebra is not unitary and associative.

EXAMPLES:

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]), Matrix([[0,1,0], [0,0,0], [0,0,0]]), Matrix([[0,0,0], [0,0,0], [0,0,1]])])
sage: B(0).characteristic_polynomial()
x^3
sage: b = B.random_element()
sage: f = b.characteristic_polynomial(); f  # random
x^3 - 8*x^2 + 16*x
sage: f(b) == 0
True
inverse()

Return the two-sided multiplicative inverse of self, if it exists.

Note

If an element of a unitary finite-dimensional algebra over a field admits a left inverse, then this is the unique left inverse, and it is also a right inverse.

EXAMPLES:

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1], [-1,0]])])
sage: C([1,2]).inverse()
1/5*e0 - 2/5*e1
is_invertible()

Return True if self has a two-sided multiplicative inverse.

Note

If an element of a unitary finite-dimensional algebra over a field admits a left inverse, then this is the unique left inverse, and it is also a right inverse.

EXAMPLES:

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1], [-1,0]])])
sage: C([1,2]).is_invertible()
True
sage: C(0).is_invertible()
False
is_nilpotent()

Return True if self is nilpotent.

EXAMPLES:

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1], [0,0]])])
sage: C([1,0]).is_nilpotent()
False
sage: C([0,1]).is_nilpotent()
True

sage: A = FiniteDimensionalAlgebra(QQ, [Matrix([0])])
sage: A([1]).is_nilpotent()
True
is_zerodivisor()

Return True if self is a left or right zero-divisor.

EXAMPLES:

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0], [0,1]]), Matrix([[0,1], [0,0]])])
sage: C([1,0]).is_zerodivisor()
False
sage: C([0,1]).is_zerodivisor()
True
left_matrix()

Return the matrix for multiplication by self from the left.

EXAMPLES:

sage: C = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,0,0], [0,0,0]]), Matrix([[0,1,0], [0,0,0], [0,0,0]]), Matrix([[0,0,0], [0,1,0], [0,0,1]])])
sage: C([1,2,0]).left_matrix()
[1 0 0]
[0 1 0]
[0 2 0]
matrix()

Return the matrix for multiplication by self from the right.

EXAMPLES:

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]), Matrix([[0,1,0], [0,0,0], [0,0,0]]), Matrix([[0,0,0], [0,0,0], [0,0,1]])])
sage: B(5).matrix()
[5 0 0]
[0 5 0]
[0 0 5]
minimal_polynomial()

Return the minimal polynomial of self.

EXAMPLES:

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]), Matrix([[0,1,0], [0,0,0], [0,0,0]]), Matrix([[0,0,0], [0,0,0], [0,0,1]])])
sage: B(0).minimal_polynomial()
x
sage: b = B.random_element()
sage: f = b.minimal_polynomial(); f  # random
x^3 + 1/2*x^2 - 7/16*x + 1/16
sage: f(b) == 0
True
vector()

Return self as a vector.

EXAMPLES:

sage: B = FiniteDimensionalAlgebra(QQ, [Matrix([[1,0,0], [0,1,0], [0,0,0]]), Matrix([[0,1,0], [0,0,0], [0,0,0]]), Matrix([[0,0,0], [0,0,0], [0,0,1]])])
sage: B(5).vector()
(5, 0, 5)

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