involution

In combinatorics, an involution is an element of order 2 in a group.

--back to combinatorics--

"Involution" can refer to a particular type of map on an algebra. If A is an algebra (roughly, a vector space equipped with a "sensible" multiplication -- see here for an axiomatic definition), then a map *: A -> A is called an involution on A when it satisfies the conditions below. Regarding the notation: these maps are usually denoted by superscripts, so that a* means "star of a", and is commonly pronounced "a star". The defining characteristics of an involution are that, for all elements a and b of A, we have:

1. (a+b)* = a* + b*

2. (ka)* = k*a*   (here k is an arbitrary scalar, so k* denotes complex conjugation)

3. (ab)* = b*a*

4. (a*)* = a

Some of the most common examples should be mentioned: the complex numbers form an algebra with involution given by the complex conjugate; the algebra of square complex matrices has an involution given by the conjugate transpose; and the adjoint map on the algebra of bounded operators on Hilbert space is an involution.

In`vo*lu"tion (?), n. [L. involutio: cf. F. involution. See Involve.]

1.

The act of involving or infolding.

2.

The state of being entangled or involved; complication; entanglement.

All things are mixed, and causes blended, by mutual involutions. Glanvill.

3.

That in which anything is involved, folded, or wrapped; envelope.

Sir T. Browne.

4. Gram.

The insertion of one or more clauses between the subject and the verb, in a way that involves or complicates the construction.

5. Math.

The act or process of raising a quantity to any power assigned; the multiplication of a quantity into itself a given number of times; -- the reverse of evolution.

6. Geom.

The relation which exists between three or more sets of points, a.a�xb7;, b.b�xb7;, c.c�xb7;, so related to a point O on the line, that the product Oa.Oa�xb7; = Ob.Ob�xb7; = Oc.Oc�xb7; is constant. Sets of lines or surfaces possessing corresponding properties may be in involution.

7. Med.

The return of an enlarged part or organ to its normal size, as of the uterus after pregnancy.

 

© Webster 1913.