Physics[Inverse] - compute the inverse of an object with respect to noncommutative products
any mathematical expression
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*B→InverseB*InverseA.
- If f is a * (commutative) product, then distribute:InverseA*B→InverseA*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
Setup(mathematicalnotation = true);
First, set prefixes for identifying anticommutative and noncommutative variables.
Setup(anticommutativeprefix = Q, noncommutativeprefix = Z);
Inverse(Z1) * Z1;
Consider now the list of objects of commutative, anticommutative, and noncommutative types.
[a, Inverse(Q), Q1 * Z2, A * B, a * (Q1 * Q2)];
The multiplicative inverses of these objects are:
In turn out that the multiplicative inverses of these inverses are the original objects themselves.
Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Physics[*], Setup, type/anticommutative, type/noncommutative
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