Elementary Operations on Subalgebras and Ideals

Bases

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) : AlgMatV -> RngIntElt

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

Basis(R) : AlgMatV -> [ AlgMatElt ]

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.

BasisElement(R, i) : AlgMatV, RngIntElt -> AlgMatElt

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

Coordinates(R, X) : AlgMatV, AlgMatVElt -> [ RngElt ]

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 U₁, ..., U_k, the coordinates are returned as the unique sequence [a₁, ..., a_k] of elements of S such that X = a₁ U₁ + ... + a_r U_r.

Intersection of Subalgebras

R meet T : AlgMat, AlgMat -> AlgMat

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.

Membership and Equality

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 notin R : AlgMatElt, AlgMat -> BoolElt

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.

R eq T : AlgMat, AlgMat -> BoolElt

Given a matrix algebra R, and a matrix algebra T, return true if R is equal to T, false otherwise.

R ne T : AlgMat, AlgMat -> BoolElt

Given a matrix algebra R and a matrix algebra T, return true if R is not equal to T, false otherwise.