Specify that the variable c commutes with all algebra elements:

Square matrices with Dot product form an algebra:

Linear endomorphisms with Composition form an algebra:

Compute with polynomials over an abstract algebra with symbolic property names:

Scalar arguments to algebra operations are interpreted as scalar multiples of the multiplicative unity:

Interpretation of unknown symbolic variables depends on the algebra settings:

When algebra generators are not specified, symbolic variables are treated as algebra elements:

Otherwise, when "ScalarVariables"Automatic, symbolic variables are treated as scalars:

Symbolic variables that are not generators or commutative variables are scalars:

When both algebra generators and scalar variables are specified, there should be no other variables:

Expressions that contain unknown symbolic variables are not elements of the algebra:

Define an algebra with generators satisfying the relation :

NonCommutativeExpand automatically reduces modulo the relation:

Define an algebra with "CommutationRelations" specified:

Algebra elements can be uniquely represented in terms of standard monomials :

Uniqueness is guaranteed by the requirement that commutation relations form a Gröbner basis:

Here "CommutationRelations" do not satisfy the required conditions:

crels do not form a Gröbner basis:

Algebra with relations crels is, in fact, isomorphic to the coefficient field:

Define an algebra with both "Relations" and "CommutationRelations" specified:

Compute the standard representation of an element:

The algebra is isomorphic to a Grassmann algebra:

Define a Weyl algebra:

Compute the standard representation of an element:

Since algebra generators are specified, symbolic variables that are not generators are treated as scalars:

Use WeylAlgebra to specify the algebra:

Define a Clifford algebra:

Compute the standard representation of an element:

Use CliffordAlgebra to specify the algebra:

Define a Grassmann algebra:

Compute the standard representation of an element:

Use GrassmannAlgebra to specify the algebra:

The order and grouping of generators determines the multi-graded monomial ordering:

NonCommutativeMonomialList lists monomials in decreasing order:

"WordOrdering" determines the ordering of monomials that differ only in variable order:

NonCommutativeMonomialList lists monomials in decreasing order:

By default, variables listed first are considered to be smaller:

NonCommutativeMonomialList lists monomials in decreasing order:

The setting also affects variables given as a function argument when "Generators" is not specified:

Check whether two noncommutative polynomials are equal:

Expand the difference of the polynomials:

Simplify an expression involving matrices and :

Find the inverses in the expression:

Generate the relations satisfied by the inverses:

Pick a variable order that makes the most complicated variables largest:

Compute the Gröbner basis of the ideal generated by the relations over the algebra of matrices:

Reducing the expression modulo the Gröbner basis shows that it is equal to the identity matrix:

Use ArraySimplify to do the above simplification steps automatically:

Prove the Woodbury matrix identity:

In the standard formulation, is an matrix, is a matrix, with , is an matrix, and is a matrix. However, by replacing , and with the block matrices , and , one may assume that all matrices belong to the algebra of matrices. It will be shown that the difference of the sides of the identity reduces to zero modulo the Gröbner basis of the ideal generated by relations implied by the properties of the matrix inverse.

Compute the difference of the sides of the Woodbury matrix identity:

Find the inverses in :

Generate the relations satisfied by the inverses:

Pick a variable order that makes the most complicated variables largest:

Compute the Gröbner basis of the ideal generated by the relations over the algebra of matrices:

Reduce modulo the Gröbner basis. The result is a zero matrix, which proves the identity:

Use ArraySimplify to do the above simplification steps automatically:

Construct a finitely presented group. The dicyclic group is given by generators and relations . The group algebra of is given by four generators:

The generators satisfy the following relations:

Compute the Gröbner basis of the ideal generated by the relations:

Note that reducing an arbitrary monomial modulo the Gröbner basis gives a monomial that does not contain and . This shows that reducing an arbitrary monomial in and of total degree yields a monomial of a total degree at most :

Hence, all elements of the group can be represented as reduced monomials of degree at most :

It has been proven that is a finite group of order :

Compute in the Clifford algebra :

Compute the standard representation of :

Show that is idempotent:

Compute the standard representation of :

Show that is nilpotent:

Compute in the Weyl algebra associated to the differential ring of bivariate polynomials:

Compute the standard representation of and :

Compute the standard representation of :

Elements of the Weyl algebra correspond to differential operators with product representing composition:

Verify that the operator corresponding to is the composition of the operators corresponding to and :