Editing Determinant (section)

== Generalizations and related notions ==

Determinants as treated above admit several variants: the [[Permanent (mathematics)|permanent]] of a matrix is defined as the determinant, except that the factors <math>\sgn(\sigma)</math> occurring in Leibniz's rule are omitted. The [[immanant of a matrix|immanant]] generalizes both by introducing a [[character theory|character]] of the [[symmetric group]] <math>S_n</math> in Leibniz's rule.

=== Determinants for finite-dimensional algebras===
For any [[associative algebra]] <math>A</math> that is [[dimension|finite-dimensional]] as a vector space over a field <math>F</math>, there is a determinant map
<ref>{{harvnb|Garibaldi|2004}}</ref>
:<math>\det : A \to F.</math>
This definition proceeds by establishing the characteristic polynomial independently of the determinant, and defining the determinant as the lowest order term of this polynomial. This general definition recovers the determinant for the [[matrix algebra]] <math>A = \operatorname{Mat}_{n \times n}(F)</math>, but also includes several further cases including the determinant of a [[quaternion]],
:<math>\det (a + ib+jc+kd) = a^2 + b^2 + c^2 + d^2</math>,
the [[Field norm|norm]] <math>N_{L/F} : L \to F</math> of a [[field extension]], as well as the [[Pfaffian]] of a skew-symmetric matrix and the [[reduced norm]] of a [[central simple algebra]], also arise as special cases of this construction.

=== Infinite matrices ===
For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated. [[Functional analysis]] provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators.

The [[Fredholm determinant]] defines the determinant for operators known as [[trace class operator]]s by an appropriate generalization of the formula
:<math>\det(I+A) = \exp(\operatorname{tr}(\log(I+A))). </math>

Another infinite-dimensional notion of determinant is the [[functional determinant]].

===Operators in von Neumann algebras===
For operators in a finite [[von Neumann algebra#Factors|factor]], one may define a positive real-valued determinant called the [[Fuglede−Kadison determinant]] using the canonical trace. In fact, corresponding to every [[State (functional analysis)#tracial state|tracial state]] on a [[von Neumann algebra]] there is a notion of Fuglede−Kadison determinant.

=== Related notions for non-commutative rings ===

For matrices over non-commutative rings, multilinearity and alternating properties are incompatible for {{math|''n'' ≥ 2}},<ref>In a non-commutative setting left-linearity (compatibility with left-multiplication by scalars) should be distinguished from right-linearity. Assuming linearity in the columns is taken to be left-linearity, one would have, for non-commuting scalars ''a'', ''b'':
<math display=block>\begin{align}
ab
  &= ab \begin{vmatrix}1&0 \\ 0&1\end{vmatrix}
   =  a \begin{vmatrix}1&0 \\ 0&b\end{vmatrix} \\[5mu]
  &=    \begin{vmatrix}a&0 \\ 0&b\end{vmatrix} 
   =  b \begin{vmatrix}a&0 \\ 0&1\end{vmatrix}
   = ba \begin{vmatrix}1&0 \\ 0&1\end{vmatrix}
   = ba,
\end{align}</math>

a contradiction. There is no useful notion of multi-linear functions over a non-commutative ring.</ref> so there is no good definition of the determinant in this setting.

For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero {{clarify span|text=bilinear form|explain=What exactly is meant by this term must be specified.  This statement is valid only if the bilinear form is required to be linear on the same side for both arguments; in contrast, Bourbaki defines a bilinear form B as having the property B(ax,yb) = aB(x,y)b, i.e., left-linear in the left argument and right-linear in the other.|date=October 2017}} with a [[Regular element (ring theory)|regular element]] of ''R'' as value on some pair of arguments implies that ''R'' is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably [[quasideterminant]]s and the [[Dieudonné determinant]]. For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include the ''q''-determinant on quantum groups, the [[Capelli determinant]] on Capelli matrices, and the [[Berezinian]] on [[supermatrices]] (i.e., matrices whose entries are elements of <math>\mathbb Z_2</math>-[[graded ring]]s).<ref>{{Citation | url = https://books.google.com/books?id=sZ1-G4hQgIIC&q=Berezinian&pg=PA116 | title = Supersymmetry for mathematicians: An introduction | isbn = 978-0-8218-3574-6 | last1 = Varadarajan | first1 = V. S | year = 2004 | publisher = American Mathematical Soc. | postscript = .}}</ref> [[Manin matrices]] form the class closest to matrices with commutative elements.