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In a similar fashion, any row or column i of F with a zero value may be eliminated if the corresponding value of x i is not desired. A reduced K may be reduced again. As a note, since each reduction requires an inversion, and each inversion is an operation with computational cost O(n 3), most large matrices are pre-processed to reduce ...
In particular, when F = C, every such character value is an algebraic integer. If F = C and χ is irreducible, then [: ()] () is an algebraic integer for all x in G. If F is algebraically closed and char(F) does not divide the order of G, then the number of irreducible characters of G is equal to the number of conjugacy classes of G.
Legendre symbol: If p is an odd prime number and a is an integer, the value of () is 1 if a is a quadratic residue modulo p; it is –1 if a is a quadratic non-residue modulo p; it is 0 if p divides a.
The characters of irreducible representations are orthogonal. The primary importance of the character group for finite abelian groups is in number theory, where it is used to construct Dirichlet characters. The character group of the cyclic group also appears in the theory of the discrete Fourier transform.
A fundamental example of a reductive group is the general linear group of invertible n × n matrices over a field k, for a natural number n. In particular, the multiplicative group G m is the group GL(1), and so its group G m (k) of k-rational points is the group k* of nonzero elements of k under
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
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A multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers. If G is any group, then the set Ch( G ) of these morphisms forms an abelian group under pointwise multiplication.