* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

For vector spaces, or more generally free modules, for which the endomorphism algebra is isomorphic to a matrix algebra, the scalar transforms are exactly the center of the endomorphism algebra, and similarly invertible transforms are the center of the general linear group GL ( V ), where they are denoted by Z ( V ), follow the usual notation for the center.

* Every endomorphism of M is either nilpotent or invertible.

Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:

If the domain is the real numbers, then each element in Y would correspond to two different elements in X (± x ), and therefore ƒ would not be invertible.

If ƒ is an invertible function with domain X and range Y, then

Specifically, if ƒ is an invertible function with domain X and range Y, then its inverse ƒ < sup >− 1 </ sup > has domain Y and range X, and the inverse of ƒ < sup >− 1 </ sup > is the original function ƒ.

* The rank of A is equal to r if and only if there exists an invertible m-by-m matrix X and an invertible n-by-n matrix Y such that

in R < nowiki ></ nowiki > X < nowiki ></ nowiki > is invertible in R < nowiki ></ nowiki > X < nowiki ></ nowiki > if and only if its constant coefficient a < sub > 0 </ sub > is invertible in R.

If R = K is a field, then a series is invertible if and only if the constant term is non-zero, i. e., if and only if the series is not divisible by X.

Any formal series with has a composition inverse provided is an invertible element of R. The coefficients are found recursively from the above formula for the coefficients of a composition, equating them with those of the composition identity X ( that is 1 at degree 1 and 0 at every degree greater than 1 ).

Since ord ( g ) = 0, the element g is invertible in K < nowiki ></ nowiki > X < nowiki ></ nowiki > ⊂ im ( D ) = ker ( Res ), whence Res ( ƒ '/ ƒ ) = m. Property ( iv ): Since ker ( Res ) ⊂ im ( D ), we can write ƒ = ƒ < sub >− 1 </ sub > X < sup >− 1 </ sup > + F ', with F ∈ K (( X )).

) Equivalently, an operator T: X → Y is Fredholm if it is invertible modulo compact operators, i. e., if there exists a bounded linear operator

In general, a line bundle ( or invertible sheaf ) on a scheme X over S is said to be very ample relative to S if there is an immersion

The differential operator is called elliptic if the element of Hom ( E < sub > x </ sub >, F < sub > x </ sub >) is invertible for all non-zero cotangent vectors at any point x of X.

If f is quasi-compact and L is an invertible sheaf on X, then L is f-ample or f-very ample if and only if its pullback L ′ is f ′- ample or f ′- very ample, respectively .< ref > EGA IV < sub > 2 </ sub >, Corollaire 2. 7. 2 .</ ref > However, it is not true that f is projective if and only if f ′ is projective.

In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of O < sub > X </ sub >- modules.

An invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of O < sub > X </ sub >- modules, that is, we have

The same definition holds in any unital ring or algebra where a is any invertible element.

To avoid attacks based on simple algebraic properties, the S-box is constructed by combining the inverse function with an invertible affine transformation.

If and are invertible then is also invertible with inverse.

The set of all invertible elements is therefore closed under multiplication and forms a Moufang loop.

For each K, the function E < sub > K </ sub >( P ) is required to be an invertible mapping on

The set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous, ( and hence homeomorphism ) so that it forms a topological group under multiplication.

The spectrum of an element x ∈ A, denoted by, consists of all those complex scalars λ such that x − λ1 is not invertible in A.

Let A be a complex unital Banach algebra in which every non-zero element x is invertible ( a division algebra ).

a − λ1 is not invertible ( because the spectrum of a is not empty ) hence a = λ1: this algebra A is naturally isomorphic to C ( the complex case of the Gelfand-Mazur theorem ).

One of the simplest examples of a category is that of groupoid, defined as a category whose arrows or morphisms are all invertible.

Furthermore, if b < sub > 1 </ sub > and b < sub > 2 </ sub > are both coprime with a, then so is their product b < sub > 1 </ sub > b < sub > 2 </ sub > ( modulo a it is a product of invertible elements, and therefore invertible ); this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.

A matrix that is not invertible has the condition number equal to infinity.

Thus for instance the determinant of a matrix with integer coefficients will be an integer, and the matrix has an inverse with integer coefficients if and only if this determinant is 1 or − 1 ( these being the only invertible elements of the integers ).

It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.

A function ƒ that has an inverse is called invertible ; the inverse function is then uniquely determined by ƒ and is denoted by ƒ < sup >− 1 </ sup > ( read f inverse, not to be confused with exponentiation ).

* The group GL < sub > n </ sub >( R ) of invertible matrices ( under matrix multiplication ) is a Lie group of dimension n < sup > 2 </ sup >, called the general linear group.

This is called the exponential map, and it maps the Lie algebra into the Lie group G. It provides a diffeomorphism between a neighborhood of 0 in and a neighborhood of e in G. This exponential map is a generalization of the exponential function for real numbers ( because R is the Lie algebra of the Lie group of positive real numbers with multiplication ), for complex numbers ( because C is the Lie algebra of the Lie group of non-zero complex numbers with multiplication ) and for matrices ( because M < sub > n </ sub >( R ) with the regular commutator is the Lie algebra of the Lie group GL < sub > n </ sub >( R ) of all invertible matrices ).

An element with a two-sided inverse in is called invertible in.

If all elements in S are invertible, S is called a loop.

In a monoid, the set of ( left and right ) invertible elements is a group, called the group of units of, and denoted by or H < sub > 1 </ sub >.

A morphism which is invertible in this sense is called an isomorphism.

A left invertible morphism is called a split mono.

In linear algebra an n-by-n ( square ) matrix A is called invertible ( some authors use nonsingular or nondegenerate ) if there exists an n-by-n matrix B such that

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. e., if there exists an invertible matrix P such that P < sup > − 1 </ sup > AP is a diagonal matrix.

* If 2 is invertible, then and are orthogonal idempotents, called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of symmetric and anti-symmetric ( Hermitian and skew Hermitian ) elements.

If A is a subalgebra of B, then for every invertible b in B the function which takes every a in A to b < sup >− 1 </ sup > a b is an algebra homomorphism ( in case, this is called an inner automorphism of B ).

A fractional ideal I is called invertible if there is another fractional ideal J such that IJ

* The set of all invertible elements in an alternative ring R forms a Moufang loop called the loop of units in R.

To make the connection to diagramatical techniques ( like Feynman diagrams ) clearer, it's often convenient to split the action S as S = 1 / 2 D < sup >− 1 </ sup >< sub > ij </ sub > φ < sup > i </ sup > φ < sup > j </ sup >+ S < sub > int </ sub > where the first term is the quadratic part and D < sup >− 1 </ sup > is an invertible symmetric ( antisymmetric for fermions ) covariant tensor of rank two in the deWitt notation whose inverse, D is called the bare propagator and S < sub > int </ sub > is the " interaction action ".

Split-complex numbers which are not invertible are called null elements.

Those operators in B ( H ) which are mapped to an invertible element of the Calkin algebra are called Fredholm operators, and their index can be described both using K-theory and directly.

A locally free sheaf ( vector bundle ) on a variety is called ample if the invertible sheaf on is ample.

In ring theory, in a given ring R any element with a multiplicative inverse is called a unit of the ring, i. e., the term may refer to any invertible element, not only the unit element 1 < sub > R </ sub >.

Two integral forms are called equivalent if there exists an invertible integral linear change of variables that transforms the first form into the second.

In mathematics, two matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that