Why $\langle w, \rho(g) v\rangle$ is called a matrix coefficient?

Question

Let $G$ be a Lie group and $H$ a Hilbert space. Let $\rho: G \to U(H)$ be a representation of $G$. $\langle w, \rho(g) v\rangle$ is called a matrix coefficient of $g$. Why $\langle w, \rho(g) v\rangle$ is called a matrix coefficient? Is $\langle w, \rho(g) v\rangle$ equal to some entry of the matrix $\rho(g)$? Thank you very much.

Answer 1

It is not a good idea to write matrices on Hilbert space) In the finite-dimensional case consider the basis that contains $w,v$. Then $\langle w, Av\rangle$ is an actual coefficient of the matrix of A. (as $\langle e_i, Ae_j\rangle =a_{ij}$)