compute the inverse of an object with respect to noncommutative products - Maple Help

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Physics[Inverse] - compute the inverse of an object with respect to noncommutative products

Calling Sequence

Inverse(f)

Parameters

f

-

any mathematical expression

Description

• 

The Inverse command, when applied to an object, represents the object's (noncommutative) multiplicative inverse; that is, Inverse(Z) * Z = Z * Inverse(Z) = 1, where * herein represents the Physics[*] product, whose commutativity depends on the operands (see also type, commutative).

• 

The %Inverse command is the inert form of Inverse; that is, it represents the same mathematical operation while displaying the operation unevaluated. To evaluate the operation, use the value command.

• 

The results returned by Inverse are constructed as follows:

  

- If f is of commutative type, then return 1f.

  

- If f is a matrix, then return its inverse.

  

- If f is equal to Inverse(g) for some g, then return g.

  

- If f is a noncommutative product, then distribute:InverseA*BInverseB*InverseA.

  

- If f is a * (commutative) product, then distribute:InverseA*BInverseA*InverseB.

  

- Otherwise, return the unevaluated expression 'Inverse'f.

• 

All noncommutative products introduced by Inverse have their operands sorted and normalized automatically by the Physics[*] operator. This ensures that the basic simplifications and identities for these products are taken into account in the returned results.

• 

A `print/Inverse` procedure makes the display of this function appear as a power, as in

Inverse(Q);

InverseQ

(1)

Examples

withPhysics:

Setupmathematicalnotation=true

mathematicalnotation=true

(2)

First, set prefixes for identifying anticommutative and noncommutative variables.

Setupanticommutativeprefix=Q,noncommutativeprefix=Z

anticommutativeprefix=Q,_λ,noncommutativeprefix=Z

(3)

InverseZ1Z1

1

(4)

Consider now the list of objects of commutative, anticommutative, and noncommutative types.

a,InverseQ,Q1Z2,AB,aQ1Q2

a,Q-1,Q1Z2,AB,aQ1Q2

(5)

The multiplicative inverses of these objects are:

mapInverse,

1a,Q,Z2-1Q1-1,1AB,Q1-1Q2-1a

(6)

In turn out that the multiplicative inverses of these inverses are the original objects themselves.

mapInverse,

a,Q-1,Q1Z2,AB,aQ1Q2

(7)

See Also

Physics, Physics conventions, Physics examples, Physics[*], Setup, type/anticommutative, type/noncommutative


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