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In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. For the case of a finite group, matrix coefficients express the action of the elements of the group in the specified representation via the entries of the corresponding matrices. Matrix coefficients of representations of Lie groups turned out to be intimately related with the theory of special functions, providing a unifying approach to large parts of this theory. Growth properties of matrix coefficients play a key role in the classification of irreducible representations of locally compact groups, in particular, reductive real and ''p''-adic groups. The formalism of matrix coefficients leads to a generalization of the notion of a modular form. In a different direction, mixing properties of certain dynamical systems are controlled by the properties of suitable matrix coefficients. == Definition == A matrix coefficient (or matrix element) of a linear representation of a group on a vector space is a function on the group, of the type : where is a vector in , is a continuous linear functional on , and is an element of . This function takes scalar values on . If is a Hilbert space, then by the Riesz representation theorem, all matrix coefficients have the form : for some vectors and in . For of finite dimension, and and taken from a standard basis, this is actually the function given by the matrix entry in a fixed place. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「matrix coefficient」の詳細全文を読む スポンサード リンク
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