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

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 $\frac{1}{f}$.
 - 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:$\mathrm{Inverse}\left(A*B\right)\to \mathrm{Inverse}\left(B\right)*\mathrm{Inverse}\left(A\right)$.
 - If $f$ is a * (commutative) product, then distribute:$\mathrm{Inverse}\left(A*B\right)\to \mathrm{Inverse}\left(A\right)*\mathrm{Inverse}\left(B\right)$.
 - Otherwise, return the unevaluated expression $'\mathrm{Inverse}'\left(f\right)$.
 • 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);
 ${\mathrm{Inverse}}{}\left({Q}\right)$ (1)

Examples

 > with(Physics):
 > Setup(mathematicalnotation = true);
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (2)

First, set prefixes for identifying anticommutative and noncommutative variables.

 > Setup(anticommutativeprefix = Q, noncommutativeprefix = Z);
 $\left[{\mathrm{anticommutativeprefix}}{=}\left\{{Q}\right\}{,}{\mathrm{noncommutativeprefix}}{=}\left\{{Z}\right\}\right]$ (3)
 > Inverse(Z1) * Z1;
 ${1}$ (4)

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

 > [a, Inverse(Q), Q1 * Z2, A * B, a * (Q1 * Q2)];
 $\left[{a}{,}{\mathrm{Inverse}}{}\left({Q}\right){,}{\mathrm{Q1}}{}{\mathrm{Z2}}{,}{A}{}{B}{,}{a}{}{\mathrm{Q1}}{}{\mathrm{Q2}}\right]$ (5)

The multiplicative inverses of these objects are:

 > map(Inverse, (5));
 $\left[\frac{{1}}{{a}}{,}{Q}{,}{\mathrm{Inverse}}{}\left({\mathrm{Z2}}\right){}{\mathrm{Inverse}}{}\left({\mathrm{Q1}}\right){,}\frac{{1}}{{A}{}{B}}{,}{-}\frac{{\mathrm{Inverse}}{}\left({\mathrm{Q1}}\right){}{\mathrm{Inverse}}{}\left({\mathrm{Q2}}\right)}{{a}}\right]$ (6)

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

 > map(Inverse, (6));
 $\left[{a}{,}{\mathrm{Inverse}}{}\left({Q}\right){,}{\mathrm{Q1}}{}{\mathrm{Z2}}{,}{A}{}{B}{,}{a}{}{\mathrm{Q1}}{}{\mathrm{Q2}}\right]$ (7)
 >