Editing Determinant (section)

== Abstract algebraic aspects {{anchor|Abstract formulation}} ==

=== Determinant of an endomorphism ===
The above identities concerning the determinant of products and inverses of matrices imply that [[matrix similarity|similar matrices]] have the same determinant: two matrices ''A'' and ''B'' are similar, if there exists an invertible matrix ''X'' such that {{math|1=''A'' = ''X''<sup>−1</sup>''BX''}}. Indeed, repeatedly applying the above identities yields

:<math>\det(A) = \det(X)^{-1} \det(B)\det(X) = \det(B) \det(X)^{-1} \det(X) = \det(B).</math>

The determinant is therefore also called a [[similarity invariance|similarity invariant]]. The determinant of a [[linear transformation]]
:<math>T : V \to V</math>
for some finite-dimensional [[vector space]] ''V'' is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of [[basis (linear algebra)|basis]] in ''V''. By the similarity invariance, this determinant is independent of the choice of the basis for ''V'' and therefore only depends on the endomorphism ''T''.

=== Square matrices over commutative rings ===

The above definition of the determinant using the Leibniz rule holds works more generally when the entries of the matrix are elements of a [[commutative ring]] <math>R</math>, such as the integers <math>\mathbf Z</math>, as opposed to the [[field (mathematics)|field]] of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies <math>\det(I) = 1</math> still holds, as do all the properties that result from that characterization.<ref>{{harvnb|Dummit|Foote|2004|loc=§11.4}}</ref>

A matrix <math>A \in \operatorname{Mat}_{n \times n}(R)</math> is invertible (in the sense that there is an inverse matrix whose entries are in <math>R</math>) if and only if its determinant is an [[Unit (ring theory)|invertible element]] in <math>R</math>.<ref>{{harvnb|Dummit|Foote|2004|loc=§11.4, Theorem 30}}</ref> For <math>R = \mathbf Z</math>, this means that the determinant is +1 or −1. Such a matrix is called [[unimodular matrix|unimodular]].

The determinant being multiplicative, it defines a [[group homomorphism]]
:<math>\operatorname{GL}_n(R) \rightarrow R^\times, </math>
between the [[general linear group]] (the group of invertible <math>n \times n</math>-matrices with entries in <math>R</math>) and the [[multiplicative group]] of units in <math>R</math>. Since it respects the multiplication in both groups, this map is a [[group homomorphism]].

[[Image:Determinant as a natural transformation.svg|300px|thumb|right|The determinant is a natural transformation.]]
Given a [[ring homomorphism]] <math>f : R \to S</math>, there is a map <math>\operatorname{GL}_n(f) : \operatorname{GL}_n(R) \to \operatorname{GL}_n(S)</math> given by replacing all entries in <math>R</math> by their images under <math>f</math>. The determinant respects these maps, i.e., the identity
:<math>f(\det((a_{i,j}))) = \det ((f(a_{i,j})))</math>
holds. In other words, the displayed commutative diagram commutes.

For example, the determinant of the [[complex conjugate]] of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo <math>m</math> of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo <math>m</math> (the latter determinant being computed using [[modular arithmetic]]). In the language of [[category theory]], the determinant is a [[natural transformation]] between the two functors <math>\operatorname{GL}_n</math> and <math>(-)^\times</math>.<ref>{{harvnb|Mac Lane|1998|loc=§I.4}}. See also ''{{section link|Natural transformation#Determinant}}''.</ref> Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of [[algebraic group]]s, from the general linear group to the [[multiplicative group]],
:<math>\det: \operatorname{GL}_n \to \mathbb G_m.</math>

=== Exterior algebra ===
{{See also|Exterior algebra#Linear algebra}}

The determinant of a linear transformation <math>T : V \to V</math> of an <math>n</math>-dimensional vector space <math>V</math> or, more generally a [[free module]] of (finite) [[rank of a module|rank]] <math>n</math> over a commutative ring <math>R</math> can be formulated in a coordinate-free manner by considering the <math>n</math>-th [[exterior algebra|exterior power]] <math>\bigwedge^n V</math> of <math>V</math>.<ref>{{harvnb|Bourbaki|1998|loc=§III.8}}</ref> The map <math>T</math> induces a linear map
:<math>\begin{align}
            \bigwedge^n T: \bigwedge^n V &\rightarrow \bigwedge^n V \\
  v_1 \wedge v_2 \wedge \dots \wedge v_n &\mapsto T v_1 \wedge T v_2 \wedge \dots \wedge T v_n.
\end{align}</math>

As <math>\bigwedge^n V</math> is one-dimensional, the map <math>\bigwedge^n T</math> is given by multiplying with some scalar, i.e., an element in <math>R</math>. Some authors such as {{harv|Bourbaki|1998}} use this fact to ''define'' the determinant to be the element in <math>R</math> satisfying the following identity (for all <math>v_i \in V</math>):
:<math>\left(\bigwedge^n T\right)\left(v_1 \wedge \dots \wedge v_n\right) = \det(T) \cdot v_1 \wedge \dots \wedge v_n.</math>

This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on <math>n</math>-tuples of vectors in <math>R^n</math>.
For this reason, the highest non-zero exterior power <math>\bigwedge^n V</math> (as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of <math>V</math> and similarly for more involved objects such as [[vector bundle]]s or [[chain complex]]es of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms <math>\bigwedge^k V</math> with <math>k < n</math>.<ref>{{harvnb|Lombardi|Quitté|2015|loc=§5.2}}, {{harvnb|Bourbaki|1998|loc=§III.5}}</ref>