Subsections

• Bases
• Intersection of Subalgebras
• Membership and Equality

The functions described here assume that the matrix algebra R is defined over a ring S with a matrix echelonization algorithm. Magma computes a basis for R considered as a S-module when necessary so then operations like membership testing can be performed. The following functions allow one to access this basis.

Dimension(R) : AlgMat -> RngIntElt

Assuming that R is a subalgebra of M_n(S), return the dimension of R, considered as a S-module.

Assuming that R is a subalgebra of M_n(S), return the S-basis of R, considered as a S-module. The basis is returned as a sequence of matrices of R.

Assuming that R is a subalgebra of M_n(S), return the i-th element of the S-basis of R. i must be between 1 and the dimension of R.

Assuming that R is a subalgebra of M_n(S), and given an element X of R, return the coordinates of X with respect to the basis of R. If R has dimension k over its coefficient ring S, and R has basis U_1, ..., U_k, the coordinates are returned as the unique sequence [a_1, ..., a_k] of elements of S such that X = a_1 U_1 + ... + a_r U_r.

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.

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.

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.