Matrix coefficient

In mathematics, a matrix coefficient of a linear representation ρ of a group G on a vector space V is any function on G of the type

L(ρ(g).v)

where v is in V, L is in the dual space of V, and g is an element of G. This function takes scalar values on G. If V is a Hilbert space, this may be written

<w,ρ(g).v>

for some w in V (Riesz representation theorem).

For V of finite dimension, and v and w taken from a standard basis, this is actually the function given by the matrix entry in a fixed place.

For Lie groups G, there is a close connection between the matrix coefficients and the theory of certain special functions