 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.Matrix coefficients of irreducible representations of finite groups play a prominent role in representation theory of these groups, as developed by Burnside, Frobenius and Schur.