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Given the matrix algebra R, defined as a subring of Mn(S), construct the
subring T of R generated by the elements specified by the list L,
where L is a list of one or more items of the following types:
- (a)
- A sequence of n2 elements of S defining an element of R;
- (b)
- An element of R;
- (c)
- A set or sequence of elements of R;
- (d)
- A subring of R;
- (e)
- A set or sequence of subrings of R.
Each element or subalgebra specified by the list must belong to the
same
complete matrix algebra. The subalgebra T will be constructed as a
subalgebra of
some matrix algebra which contains each of the elements and subalgebras
specified in the list.
The generators of T consist of the elements specified by the terms of the
list L together with the stored generators for subalgebras specified by terms
of the list. Repetitions of an element and occurrences of the identity
element are removed (unless T is trivial).
The constructor returns the subalgebra T and the inclusion homomorphism
f : T -> R.
Given the matrix algebra R, construct the two-sided ideal I of R
generated by the elements of R specified by the list L, where the
possibilities for L are the same as for the sub-constructor.
Given the matrix algebra R, construct the left ideal I of R
generated by the elements of R specified by the list L, where the
possibilities for L are the same as for the sub-constructor.
Given the matrix algebra R, construct the right ideal I of R generated by
the elements of R specified in the list L, where the possibilities for L
are the same as for the sub-constructor.
We construct the subalgebra of the matrix algebra A (defined
above) that is generated by the first generator.
> Q := RationalField();
> A := MatrixAlgebra< Q, 3 | [ 1/3,0,0, 3/2,3,0, -1/2,4,3],
> [ 3,0,0, 1/2,-5,0, 8,-1/2,4] >;
> B := sub< A | A.1 >;
> Dimension(B);
3
> B: Maximal;
Matrix Algebra of degree 3 and dimension 3 with 1 generator
over Rational Field
Generators:
[ 1/3 0 0]
[ 3/2 3 0]
[-1/2 4 3]
Basis:
[1 0 0]
[0 1 0]
[0 0 1]
[ 0 0 0]
[ 1 16/9 0]
[ 0 88/27 16/9]
[ 0 0 0]
[ 0 0 0]
[ 1 16/9 0]
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