Dieudonné determinant
Encyclopedia
In linear algebra
, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division ring
s.
It was introduced by .
If K is a division ring, then the Dieudonné determinant is a homomorphism
of groups from the group GLn(K) of invertible n by n matrices over K onto the abelianization K*/[K*, K*] of the multiplicative group K* of K.
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the Dieudonné determinant is a generalization of the determinant of a matrix to matrices over division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
s.
It was introduced by .
If K is a division ring, then the Dieudonné determinant is a homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
of groups from the group GLn(K) of invertible n by n matrices over K onto the abelianization K*/[K*, K*] of the multiplicative group K* of K.