In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the concept of an inverse element generalises the concepts of
opposite () and
reciprocal () of numbers.
Given an
operation denoted here , and an
identity element denoted , if , one says that is a left inverse of , and that is a right inverse of . (An identity element is an element such that and for all and for which the left-hand sides are defined.)
When the operation is
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
, if an element has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the ''inverse element'' or simply the ''inverse''. Often an adjective is added for specifying the operation, such as in
additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
,
multiplicative inverse, and
functional inverse. In this case (associative operation), an invertible element is an element that has an inverse.
Inverses are commonly used in
groupswhere every element is invertible, and
ringswhere invertible elements are also called
units. They are also commonly used for operations that are not defined for all possible operands, such as
inverse matrices and
inverse functions. This has been generalized to
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, where, by definition, an
isomorphism is an invertible
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
.
The word 'inverse' is derived from la,
inversus that means 'turned upside down', 'overturned'. This may take its origin from the case of
fractions, where the (multiplicative) inverse is obtained by exchanging the numerator and the denominator (the inverse of
is
).
Definitions and basic properties
The concepts of ''inverse element'' and ''invertible element'' are commonly defined for
binary operations that are everywhere defined (that is, the operation is defined for any two elements of its
domain). However, these concepts are commonly used with
partial operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary ope ...
s, that is operations that are not defined everywhere. Common examples are
matrix multiplication,
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
and composition of
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s in a
category. It follows that the common definitions of
associativity and
identity element must be extended to partial operations; this is the object of the first subsections.
In this section, is a
set (possibly a
proper class
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
) on which a partial operation (possibly total) is defined, which is denoted with
Associativity
A partial operation is
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
if
:
for every in for which one of the members of the equality is defined; the equality means that the other member of the equality must also be defined.
Examples of non-total associative operations are
multiplication of matrices of arbitrary size, and
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
.
Identity elements
Let
be a possibly
partial associative operation on a set .
An ''
identity element'', or simply an ''identity'' is an element such that
:
for every and for which the left-hand sides of the equalities are defined.
If and are two identity elements such that
is defined, then
(This results immediately from the definition, by
)
It follows that a total operation has at most one identity element, and if and are different identities, then
is not defined.
For example, in the case of
matrix multiplication, there is one
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
for every positive integer , and two identity matrices of different size cannot be multiplied together.
Similarly,
identity functions are identity elements for
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
, and the composition of the identity functions of two different sets are not defined.
Left and right inverses
If
where is an identity element, one says that is a ''left inverse'' of , and is a ''right inverse'' of .
Left and right inverses do not always exist, even when the operation is total and associative. For example, addition is a total associative operation on
nonnegative integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
s, which has as
additive identity, and is the only element that has an
additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
. This lack of inverses is the main motivation for extending the
natural numbers into the integers.
An element can have several left inverses and several right inverses, even when the operation is total and associative. For example, consider the
functions from the integers to the integers. The ''doubling function''
has infinitely many left inverses under
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
, which are the functions that divide by two the even numbers, and give any value to odd numbers. Similarly, every function that maps to either
or
is a right inverse of the function
the
floor function that maps to
or
depending whether is even or odd.
More generally, a function has a left inverse for
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
if and only if it is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
, and it has a right inverse if and only if it is
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
.
In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, right inverses are also called
sections, and left inverses are called
retractions.
Inverses
An element is ''invertible'' under an operation if it has a left inverse and a right inverse.
In the common case where the operation is associative, the left and right inverse of an element are equal and unique. Indeed, if and are respectively a left inverse and a right inverse of , then
:
''The inverse'' of an invertible element is its unique left or right inverse.
If the operation is denoted as an addition, the inverse, or
additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
, of an element is denoted
Otherwise, the inverse of is generally denoted
or, in the case of a
commutative multiplication
When there may be a confusion between several operations, the symbol of the operation may be added before the exponent, such as in
The notation
is not commonly used for
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
, since
can be used for the
multiplicative inverse.
If and are invertible, and
is defined, then
is invertible, and its inverse is
An invertible
homomorphism is called an
isomorphism. In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, an invertible
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
is also called an
isomorphism.
In groups
A
group is a
set with an
associative operation that has an identity element, and for which every element has an inverse.
Thus, the inverse is a
function from the group to itself that may also be considered as an operation of
arity
Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. In ...
one. It is also an
involution, since the inverse of the inverse of an element is the element itself.
A group may
act on a set as
transformations of this set. In this case, the inverse
of a group element
defines a transformation that is the inverse of the transformation defined by
that is, the transformation that "undoes" the transformation defined by
For example, the
Rubik's cube group represents the finite sequences of elementary moves. The inverse of such a sequence is obtained by applying the inverse of each move in the reverse order.
In monoids
A
monoid is a set with an
associative operation that has an
identity element.
The ''invertible elements'' in a monoid form a
group under monoid operation.
A
ring is a monoid for ring multiplication. In this case, the invertible elements are also called
units and form the
group of units
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the element is unique for this ...
of the ring.
If a monoid is not
commutative, there may exist non-invertible elements that have a left inverse or a right inverse (not both, as, otherwise, the element would be invertible).
For example, the set of the
functions from a set to itself is a monoid under
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
. In this monoid, the invertible elements are the
bijective functions; the elements that have left inverses are the
injective function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
s, and those that have right inverses are the
surjective functions.
Given a monoid, one may want extend it by adding inverse to some elements. This is generally impossible for non-commutative monoids, but, in a commutative monoid, it is possible to add inverses to the elements that have the
cancellation property (an element has the cancellation property if
implies
and
implies This extension of a monoid is allowed by
Grothendieck group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic i ...
construction. This is the method that is commonly used for constructing
integers from
natural numbers,
rational numbers from
integers and, more generally, the
field of fractions of an
integral domain, and
localizations of
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
s.
In rings
A
ring is an
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
with two operations, ''addition'' and ''multiplication'', which are denoted as the usual operations on numbers.
Under addition, a ring is an
abelian group, which means that addition is
commutative and
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
; it has an identity, called the
additive identity, and denoted ; and every element has an inverse, called its
additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
and denoted . Because of commutativity, the concepts of left and right inverses are meaningless since they do not differ from inverses.
Under multiplication, a ring is a
monoid; this means that multiplication is associative and has an identity called the
multiplicative identity
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
and denoted . An ''invertible element '' for multiplication is called a
unit. The inverse or
multiplicative inverse (for avoiding confusion with additive inverses) of a unit is denoted
or, when the multiplication is commutative,
The additive identity is never a unit, except when the ring is the
zero ring, which has as its unique element.
If is the only non-unit, the ring is a
field if the multiplication is commutative, or a
division ring otherwise.
In a
noncommutative ring (that is, a ring whose multiplication is not commutative), a non-invertible element may have one or several left or right inverses. This is, for example, the case of the functions from the integers to themselves, which form a ring for
pointwise operation In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
s; see above, '.
A
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
(that is, a ring whose multiplication is commutative) may be extended by adding inverses to elements that are not
zero divisors (that is, their product with a nonzero element cannot be ). This is the process of
localization, which produces, in particular, the field of
rational numbers from the ring of integers, and, more generally, the
field of fractions of an
integral domain. Localization is also used with zero divisors, but, in this case the original ring is not a
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those wh ...
of the localisation; instead, it is mapped non-injectively to the localization.
Matrices
Matrix multiplication is commonly defined for
matrices over a
field, and straightforwardly extended to matrices over
rings,
rngs and
semirings. However, ''in this section, only matrices over a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
are considered'', because of the use of the concept of
rank and
determinant.
If is a matrix (that is, a matrix with rows and columns), and is a matrix, the product is defined if , and only in this case. An
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
, that is, an identity element for matrix multiplication is a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
(same number for rows and columns) whose entries of the
main diagonal are all equal to , and all other entries are .
An
invertible matrix is an invertible element under matrix multiplication. A matrix over a commutative ring is invertible if and only if its determinant is a
unit in (that is, is invertible in . In this case, its
inverse matrix can be computed with
Cramer's rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants o ...
.
If is a field, the determinant is invertible if and only if it is not zero. As the case of fields is more common, one see often invertible matrices defined as matrices with a nonzero determinant, but this is incorrect over rings.
In the case of
integer matrices In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. In ...
(that is, matrices with integer entries), an invertible matrix is a matrix that has an inverse that is also an integer matrix. Such a matrix is called a
unimodular matrix for distinguishing it from matrices that are invertible over the
real numbers. A square integer matrix is unimodular if and only if its determinant is or , since these two numbers are the only units in the ring of integers.
A matrix has a left inverse if and only if its rank equals its number of columns. This left inverse is not unique except for square matrices where the left inverse equal the inverse matrix. Similarly, a right inverse exists if and only if the rank equals the number of rows; it is not unique in the case of a rectangular matrix, and equals the inverse matrix in the case of a square matrix.
Functions, homomorphisms and morphisms
Composition is a
partial operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary ope ...
that generalizes to
homomorphisms of
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s and
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s of
categories into operations that are also called ''composition'', and share many properties with function composition.
In all the case, composition is
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
.
If
and
the composition
is defined if and only if
or, in the function and homomorphism cases,
In the function and homomorphism cases, this means that the
codomain of
equals or is included in the
domain of . In the morphism case, this means that the
codomain of
equals the
domain of .
There is an ''identity''
for every object (
set, algebraic structure or
object), which is called also an
identity function in the function case.
A function is invertible if and only if it is a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
. An invertible homomorphism or morphism is called an isomorphism. An homomorphism of algebraic structures is an isomorphism if and only if it is a bijection. The inverse of a bijection is called an
inverse function. In the other cases, one talks of ''inverse isomorphisms''.
A function has a left inverse or a right inverse if and only it is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
or
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
, respectively. An homomorphism of algebraic structures that has a left inverse or a right inverse is respectively injective or surjective, but the converse is not true in some algebraic structures. For example, the converse is true for
vector spaces but not for
modules over a ring: a homomorphism of modules that has a left inverse of a right inverse is called respectively a
split epimorphism or a
split monomorphism
In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism.
In other words, if f: X\to Y and g: Y\to X are morphisms whose composition f \circ g: Y\to Y is t ...
. This terminology is also used for morphisms in any category.
Generalizations
In a unital magma
Let
be a unital
magma, that is, a
set with a
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
and an
identity element . If, for
, we have
, then
is called a left inverse of
and
is called a right inverse of
. If an element
is both a left inverse and a right inverse of
, then
is called a two-sided inverse, or simply an inverse, of
. An element with a two-sided inverse in
is called invertible in
. An element with an inverse element only on one side is left invertible or right invertible.
Elements of a unital magma
may have multiple left, right or two-sided inverses. For example, in the magma given by the Cayley table
the elements 2 and 3 each have two two-sided inverses.
A unital magma in which all elements are invertible need not be a
loop. For example, in the magma
given by the
Cayley table
every element has a unique two-sided inverse (namely itself), but
is not a loop because the Cayley table is not a
Latin square.
Similarly, a loop need not have two-sided inverses. For example, in the loop given by the Cayley table
the only element with a two-sided inverse is the identity element 1.
If the operation
is
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
then if an element has both a left inverse and a right inverse, they are equal. In other words, in a
monoid (an associative unital magma) every element has at most one inverse (as defined in this section). In a monoid, the set of invertible elements is a
group, called the
group of units
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the element is unique for this ...
of
, and denoted by
or ''H''
1.
In a semigroup
The definition in the previous section generalizes the notion of inverse in group relative to the notion of identity. It's also possible, albeit less obvious, to generalize the notion of an inverse by dropping the identity element but keeping associativity; that is, in a
semigroup.
In a semigroup ''S'' an element ''x'' is called (von Neumann) regular if there exists some element ''z'' in ''S'' such that ''xzx'' = ''x''; ''z'' is sometimes called a ''
pseudoinverse''. An element ''y'' is called (simply) an inverse of ''x'' if ''xyx'' = ''x'' and ''y'' = ''yxy''. Every regular element has at least one inverse: if ''x'' = ''xzx'' then it is easy to verify that ''y'' = ''zxz'' is an inverse of ''x'' as defined in this section. Another easy to prove fact: if ''y'' is an inverse of ''x'' then ''e'' = ''xy'' and ''f'' = ''yx'' are
idempotents, that is ''ee'' = ''e'' and ''ff'' = ''f''. Thus, every pair of (mutually) inverse elements gives rise to two idempotents, and ''ex'' = ''xf'' = ''x'', ''ye'' = ''fy'' = ''y'', and ''e'' acts as a left identity on ''x'', while ''f'' acts a right identity, and the left/right roles are reversed for ''y''. This simple observation can be generalized using
Green's relations: every idempotent ''e'' in an arbitrary semigroup is a left identity for ''R
e'' and right identity for ''L
e''. An intuitive description of this fact is that every pair of mutually inverse elements produces a local left identity, and respectively, a local right identity.
In a monoid, the notion of inverse as defined in the previous section is strictly narrower than the definition given in this section. Only elements in the Green class
''H''1 have an inverse from the unital magma perspective, whereas for any idempotent ''e'', the elements of ''H''
e have an inverse as defined in this section. Under this more general definition, inverses need not be unique (or exist) in an arbitrary semigroup or monoid. If all elements are regular, then the semigroup (or monoid) is called regular, and every element has at least one inverse. If every element has exactly one inverse as defined in this section, then the semigroup is called an
inverse semigroup. Finally, an inverse semigroup with only one idempotent is a group. An inverse semigroup may have an
absorbing element 0 because 000 = 0, whereas a group may not.
Outside semigroup theory, a unique inverse as defined in this section is sometimes called a quasi-inverse. This is generally justified because in most applications (for example, all examples in this article) associativity holds, which makes this notion a generalization of the left/right inverse relative to an identity (see
Generalized inverse).
''U''-semigroups
A natural generalization of the inverse semigroup is to define an (arbitrary) unary operation ° such that (''a''°)° = ''a'' for all ''a'' in ''S''; this endows ''S'' with a type 2,1 algebra. A semigroup endowed with such an operation is called a ''U''-semigroup. Although it may seem that ''a''° will be the inverse of ''a'', this is not necessarily the case. In order to obtain interesting notion(s), the unary operation must somehow interact with the semigroup operation. Two classes of ''U''-semigroups have been studied:
* ''I''-semigroups, in which the interaction axiom is ''aa''°''a'' = ''a''
*
*-semigroups, in which the interaction axiom is (''ab'')° = ''b''°''a''°. Such an operation is called an
involution, and typically denoted by ''a''*
Clearly a group is both an ''I''-semigroup and a *-semigroup. A class of semigroups important in semigroup theory are
completely regular semigroups; these are ''I''-semigroups in which one additionally has ''aa''° = ''a''°''a''; in other words every element has commuting pseudoinverse ''a''°. There are few concrete examples of such semigroups however; most are
completely simple semigroups. In contrast, a subclass of *-semigroups, the
*-regular semigroups (in the sense of Drazin), yield one of best known examples of a (unique) pseudoinverse, the
Moore–Penrose inverse. In this case however the involution ''a''* is not the pseudoinverse. Rather, the pseudoinverse of ''x'' is the unique element ''y'' such that ''xyx'' = ''x'', ''yxy'' = ''y'', (''xy'')* = ''xy'', (''yx'')* = ''yx''. Since *-regular semigroups generalize inverse semigroups, the unique element defined this way in a *-regular semigroup is called the ''generalized inverse'' or ''Moore–Penrose inverse''.
Semirings
Examples
All examples in this section involve associative operators.
Galois connections
The lower and upper adjoints in a (monotone)
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the funda ...
, ''L'' and ''G'' are quasi-inverses of each other; that is, ''LGL'' = ''L'' and ''GLG'' = ''G'' and one uniquely determines the other. They are not left or right inverses of each other however.
Generalized inverses of matrices
A
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
with entries in a
field 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
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
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
is invertible
if and only if its determinant is invertible in
.
Non-square matrices of
full rank
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dime ...
have several one-sided inverses:
* For
we have left inverses; for example,
* For