Inverse element

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Categories: Algebra | Abstract algebra | Binary operations

In mathematics, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element.

1 Formal definition
2 Calculation
3 Example
4 See also
5 External links

Formal definition

Let Failed to parse (Missing texvc executable; please see math/README to configure.): S

be a set with a binary operation Failed to parse (Missing texvc executable; please see math/README to configure.): *

. If Failed to parse (Missing texvc executable; please see math/README to configure.): e

is an identity element of Failed to parse (Missing texvc executable; please see math/README to configure.): (S,*)
and Failed to parse (Missing texvc executable; please see math/README to configure.): a*b=e

, then Failed to parse (Missing texvc executable; please see math/README to configure.): a

is called a left inverse of Failed to parse (Missing texvc executable; please see math/README to configure.): b
and Failed to parse (Missing texvc executable; please see math/README to configure.): b
is called a right inverse of Failed to parse (Missing texvc executable; please see math/README to configure.): a

. If an element Failed to parse (Missing texvc executable; please see math/README to configure.): x

is both a left inverse and a right inverse of Failed to parse (Missing texvc executable; please see math/README to configure.): y

, then Failed to parse (Missing texvc executable; please see math/README to configure.): x

is called a two-sided inverse, or simply an inverse, of Failed to parse (Missing texvc executable; please see math/README to configure.): y

. An element with a two-sided inverse in Failed to parse (Missing texvc executable; please see math/README to configure.): S

is called invertible in Failed to parse (Missing texvc executable; please see math/README to configure.): S

. An element with an inverse element only on one side is left invertible, resp. right invertible.

Just like Failed to parse (Missing texvc executable; please see math/README to configure.): (S,*)

can have several left identities or several right identities, it is possible for an element to have several left inverses or several right inverses (but note that their definition above uses a two-sided identity Failed to parse (Missing texvc executable; please see math/README to configure.): e

). It can even have several left inverses and several right inverses.

If the operation Failed to parse (Missing texvc executable; please see math/README to configure.): *

is associative then if an element has both a left inverse and a right inverse, they are equal and unique. In this case, the set of (left and right) invertible elements is a group, called the group of units of Failed to parse (Missing texvc executable; please see math/README to configure.): S

, and denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): U(S)

or Failed to parse (Missing texvc executable; please see math/README to configure.): S^*

Calculation

Every real number Failed to parse (Missing texvc executable; please see math/README to configure.): x

has an additive inverse (i.e. an inverse with respect to addition) given by Failed to parse (Missing texvc executable; please see math/README to configure.): -x

. Every nonzero real number Failed to parse (Missing texvc executable; please see math/README to configure.): x

has a multiplicative inverse (i.e. an inverse with respect to multiplication) given by Failed to parse (Missing texvc executable; please see math/README to configure.): \frac 1{x}

. By contrast, zero has no multiplicative inverse.

A square matrix Failed to parse (Missing texvc executable; please see math/README to configure.): M

with entries in a field Failed to parse (Missing texvc executable; please see math/README to configure.): K
is invertible (in the set of all square matrices of the same size, under matrix multiplication) if and only if its determinant is different from zero. If the determinant of Failed to parse (Missing texvc executable; please see math/README to configure.): M
is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one. See invertible matrix for more.

More generally, a square matrix over a commutative ring Failed to parse (Missing texvc executable; please see math/README to configure.): R

is invertible if and only if its determinant is invertible in Failed to parse (Missing texvc executable; please see math/README to configure.): R

A function Failed to parse (Missing texvc executable; please see math/README to configure.): g

is the left (resp. right) inverse of a function Failed to parse (Missing texvc executable; please see math/README to configure.): f
(for function composition), if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): g o f
(resp. Failed to parse (Missing texvc executable; please see math/README to configure.): f o g

) is the identity function on the domain (resp. codomain) of Failed to parse (Missing texvc executable; please see math/README to configure.): f .

For Failed to parse (Missing texvc executable; please see math/README to configure.): A:m\times n \mid m>n

we have a left inverse: Failed to parse (Missing texvc executable; please see math/README to configure.):  \underbrace{ (A^{T}A)^{-1}A^{T} }_{ A^{-1}_{left} } A = I_{n}

For Failed to parse (Missing texvc executable; please see math/README to configure.): A:m\times n \mid m<n

we have a right inverse: Failed to parse (Missing texvc executable; please see math/README to configure.):  A \underbrace{ A^{T}(AA^{T})^{-1} }_{ A^{-1}_{right} } = I_{m}

^[1]

Example

Failed to parse (Missing texvc executable; please see math/README to configure.): A:2\times 3 = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}

So, as m<n, we have a right inverse. Failed to parse (Missing texvc executable; please see math/README to configure.): A^{-1}_{right} = A^{T}(AA^{T})^{-1}

Failed to parse (Missing texvc executable; please see math/README to configure.): AA^{T} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\cdot \begin{bmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{bmatrix} = \begin{bmatrix} 14 & 32\\ 32 & 77 \end{bmatrix}
Failed to parse (Missing texvc executable; please see math/README to configure.): (AA^{T})^{-1} = \begin{bmatrix} 14 & 32\\ 32 & 77 \end{bmatrix}^{-1} = \frac{1}{54} \begin{bmatrix} 77 & -32\\ -32 & 14 \end{bmatrix}

Failed to parse (Missing texvc executable; please see math/README to configure.): A^{T}(AA^{T})^{-1} = \frac{1}{54}\begin{bmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{bmatrix} \cdot \begin{bmatrix} 77 & -32\\ -32 & 14 \end{bmatrix} = \frac{1}{18} \begin{bmatrix} -17 & 8\\ -2 & 2\\ 13 & -4 \end{bmatrix} = A^{-1}_{right}

The left inverse doesn't exist, because Failed to parse (Missing texvc executable; please see math/README to configure.): A^{T}A = \begin{bmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} 17 & 22 & 27 \\ 22 & 29 & 36\\ 27 & 36 & 45 \end{bmatrix}

Is a singular matrix, and can't be inverted.