Subsections

• Rank
• Intersection of Subalgebras
• Membership and Equality

The operations described here assume that the matrix algebra is defined over a principal ideal ring.

Dimension(R) : AlgMat -> RngIntElt

Assume that R is a subalgebra of M_n(K), where K is a field. This function returns the dimension of R, considered as a K-module.

Given algebras R and S that are subalgebras of the same complete algebra M_n(S), where S is a PIR, this operator constructs their intersection.

The operations described here assume that the matrix algebra is defined over a principal ideal ring.

x in R : AlgMatElt, AlgMat -> BoolElt

X subset R : { AlgMatElt } , AlgMat -> BoolElt

T subset R : AlgMat, AlgMat -> BoolElt

Given a matrix x (set of matrices X, matrix algebra T) and a matrix algebra R all belonging to a common matrix algebra defined over a PIR, return true if x (X, T, respectively) is contained in R, false otherwise.

X notsubset R : { AlgMatElt } , AlgMat -> BoolElt

T notsubset R : AlgMat, AlgMat -> BoolElt

Given a matrix x (set of matrices X, matrix algebra T) and a matrix algebra R all belonging to a common matrix algebra defined over a PIR, return true if x (X, T, respectively) is not contained in R, false otherwise.

[Future release] X subset I : { AlgMatElt } , AlgMatIdl -> BoolElt

[Future release] J subset I : AlgMatIdl, AlgMatIdl -> BoolElt

Given a matrix x (set of matrices X, matrix algebra ideal J) and an ideal I all belonging to a common matrix algebra defined over a PIR, return true if x (respectively, X, J) is contained in I, false otherwise.

[Future release] X notsubset I : { AlgMatElt } , AlgMatIdl -> BoolElt

[Future release] J notsubset I : AlgMatIdl, AlgMatIdl -> BoolElt

Given a matrix x (set of matrices X, matrix algebra ideal J) and an ideal I all belonging to a common matrix algebra defined over a PIR, return true if x (respectively, X, J) is not contained in I, false otherwise.

[Future release] I eq J : AlgMatIdl, AlgMatIdl -> BoolElt

Given a matrix algebra R (respectively, ideal I belonging to a matrix algebra R), and a matrix algebra T, (respectively, ideal J), return true if R (respectively, I) is equal to T ( respectively, J), false otherwise.

[Future release] I ne J : AlgMatIdl, AlgMatIdl -> BoolElt

Given a matrix algebra R (respectively, ideal I belonging to a matrix algebra R), and a matrix algebra T, (respectively, ideal J), return true if R (respectively, I) is not equal to T ( respectively, J), false otherwise.