Given two matrix algebras R and T, where R and T have the same
coefficient ring S, return the direct sum D of R and T
(with the action given by the direct sum of the action of R
and the action of T).
Given two unital matrix algebras A and B, where A and B have the same
coefficient ring S, construct the tensor product of A and B.
We construct the direct product and tensor product of the
matrix algebra A (defined above) with itself.
> 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] >;
> AplusA := DirectSum(A, A);
> AplusA: Maximal;
Matrix Algebra of degree 6 with 4 generators over
Rational Field
Generators:
[ 1/3 0 0 0 0 0]
[ 3/2 3 0 0 0 0]
[-1/2 4 3 0 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 0 0 0]
[ 3 0 0 0 0 0]
[ 1/2 -5 0 0 0 0]
[ 8 -1/2 4 0 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 1/3 0 0]
[ 0 0 0 3/2 3 0]
[ 0 0 0 -1/2 4 3]
[ 0 0 0 0 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 3 0 0]
[ 0 0 0 1/2 -5 0]
[ 0 0 0 8 -1/2 4]
> AtimesA := TensorProduct(A, A);
> AtimesA: Maximal;
Matrix Algebra of degree 9 with 4 generators over
Rational Field
Generators:
[ 1/3 0 0 0 0 0 0 0 0]
[ 0 1/3 0 0 0 0 0 0 0]
[ 0 0 1/3 0 0 0 0 0 0]
[ 3/2 0 0 3 0 0 0 0 0]
[ 0 3/2 0 0 3 0 0 0 0]
[ 0 0 3/2 0 0 3 0 0 0]
[-1/2 0 0 4 0 0 3 0 0]
[ 0 -1/2 0 0 4 0 0 3 0]
[ 0 0 -1/2 0 0 4 0 0 3]
[ 3 0 0 0 0 0 0 0 0]
[ 0 3 0 0 0 0 0 0 0]
[ 0 0 3 0 0 0 0 0 0]
[ 1/2 0 0 -5 0 0 0 0 0]
[ 0 1/2 0 0 -5 0 0 0 0]
[ 0 0 1/2 0 0 -5 0 0 0]
[ 8 0 0 -1/2 0 0 4 0 0]
[ 0 8 0 0 -1/2 0 0 4 0]
[ 0 0 8 0 0 -1/2 0 0 4]
[ 1/3 0 0 0 0 0 0 0 0]
[ 3/2 3 0 0 0 0 0 0 0]
[-1/2 4 3 0 0 0 0 0 0]
[ 0 0 0 1/3 0 0 0 0 0]
[ 0 0 0 3/2 3 0 0 0 0]
[ 0 0 0 -1/2 4 3 0 0 0]
[ 0 0 0 0 0 0 1/3 0 0]
[ 0 0 0 0 0 0 3/2 3 0]
[ 0 0 0 0 0 0 -1/2 4 3]
[ 3 0 0 0 0 0 0 0 0]
[ 1/2 -5 0 0 0 0 0 0 0]
[ 8 -1/2 4 0 0 0 0 0 0]
[ 0 0 0 3 0 0 0 0 0]
[ 0 0 0 1/2 -5 0 0 0 0]
[ 0 0 0 8 -1/2 4 0 0 0]
[ 0 0 0 0 0 0 3 0 0]
[ 0 0 0 0 0 0 1/2 -5 0]
[ 0 0 0 0 0 0 8 -1/2 4]
Given an element a of the matrix algebra Q and an element b of the matrix
algebra R, form the direct sum of matrices a and b. The square is returned
as an element of the matrix algebra T, which must be the direct sum of the
parent of a and the parent of b.
Given an element a of the matrix algebra Mn(S), form the exterior
square of a as an element of Mm(S), where m = n(n - 1)/2.
Given an element a of the matrix algebra Mn(S), form the rth exterior
power of a as an element of Mm(S), where m = n choose r.
Given an element a of the matrix algebra Mn(S), form the symmetric
square of a as an element of Mm(S), where m = n(n + 1)/2.
Given an element a of the matrix algebra Mn(S), form the rth symmetric
power of a as an element of Mm(S), for the appropriate m.
Given an element a belonging to a subalgebra of Mn1(S) and an
element b belonging to a subalgebra of Mn2(S), construct the
tensor product of a and b as an element of the matrix algebra Mn(S),
where n = n1 * n2.
V2.28, 13 July 2023