invertierbare

Invertierbare is a German adjective used in mathematical terminology to describe an object that possesses an inverse with respect to a given operation. The corresponding noun form is invertierbarkeit or the phrase “invertierbare …” when describing specific objects such as matrices or functions.

In linear algebra, a square matrix A is invertible if there exists a matrix B such that AB = BA = I, where I is the identity matrix. Invertibility is equivalent to having a nonzero determinant and full rank; for matrices over a field, these conditions all coincide. An invertible matrix is also called nonsingular and has a unique inverse.

In the context of functions, a function f: X → Y is invertible if it is bijective, meaning it is both injective and surjective. In this case there exists an inverse function f⁻¹: Y → X, which reverses the action of f and satisfies f⁻¹(f(x)) = x for all x in X and f(f⁻¹(y)) = y for all y in Y.

In ring theory and abstract algebra, an element is invertible if there exists another element that multiplies with it to yield the multiplicative identity. Such elements are called units. In a group, every element is invertible by definition; in more general structures, invertibility can be a distinguishing property of certain elements or morphisms.

Invertibility can be left- or right-invertible in non-commutative settings, and not all objects are invertible. The concept is central to solving equations, constructing inverse mappings, and understanding structural properties across algebra, analysis, and category theory.